Morphology of three-body quantum states from machine learning

David HuberTechnische Universitat Darmstadt, Department of Physics,

Institut fur Kernphysik, 64289 Darmstadt, Germany

Oleksandr V. MarchukovTechnische Universitat Darmstadt, Institut fur Angewandte Physik, Hochschulstraße 4a, 64289 Darmstadt, Germany

Hans-Werner HammerTechnische Universitat Darmstadt, Department of Physics,Institut fur Kernphysik, 64289 Darmstadt, Germany and

ExtreMe Matter Institute EMMI and Helmholtz Forschungsakademie Hessen fur FAIR (HFHF),GSI Helmholtzzentrum fur Schwerionenforschung GmbH, 64291 Darmstadt, Germany

Artem G. VolosnievInstitute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria

The relative motion of three impenetrable particles on a ring, in our case two identical fermionsand one impurity, is isomorphic to a triangular quantum billiard. Depending on the ratio κ of theimpurity and fermion masses, the billiards can be integrable or non-integrable (also referred to inthe main text as chaotic). To set the stage, we first investigate the energy level distributions of thebilliards as a function of 1/κ ∈ [0, 1] and find no evidence of integrable cases beyond the limitingvalues 1/κ = 1 and 1/κ = 0. Then, we use machine learning tools to analyze properties of probabilitydistributions of individual quantum states. We find that convolutional neural networks can correctlyclassify integrable and non-integrable states. The decisive features of the wave functions are thenormalization and a large number of zero elements, corresponding to the existence of a nodal line.The network achieves typical accuracies of 97%, suggesting that machine learning tools can be usedto analyze and classify the morphology of probability densities obtained in theory or experiment.

I. INTRODUCTION

The correspondence principle conjectures that highlyexcited states of a quantum system carry informationabout the classical limit [1]. In particular, it implies thatthere must be means to tell a difference between a ‘typ-ical’ high-energy quantum state that corresponds to anintegrable classical system from a ‘typical’ high-energystate that corresponds to a chaotic system. A discoveryof such means is a complicated task that requires a co-herent effort of physicists, mathematicians, and philoso-phers [2–4]. Currently, there are two main approachesto study chaotic features in quantum mechanics. Oneapproach relies on the statistical analysis of the energylevels of a quantum-mechanical system. Another focuseson the morphology of wave functions. These approachesled to a few celebrated conjectures that postulate featuresof energy spectra and properties of eigenstates [5–7]. Thepostulates are widely accepted now, thanks to numericalas well as experimental data [8, 9].

Numerical and experimental data sets produced toconfirm the proposed conjectures are so large that it isdifficult, if not hopeless, for the human eye to find univer-sal patterns beyond what has been conjectured. There-fore, it is logical to look for computational tools thatcan learn (with or without supervision) universal pat-terns from large datasets. One such tool is deep learn-ing (DL) [10], which is a machine learning method thatuses artificial neural networks with multiple layers for a

progressive learning of features from the input data. Itrequires very little engineering by hand, and can easilybe used to analyze big data across disciplines, in partic-ular in physics [11]. DL tools present an opportunity togo beyond the standard approaches of quantum chaolo-gists [12]. For example, in this paper, neural networksbuilt upon many states are used to analyze the morphol-ogy of individual wave functions. Therefore, DL providesus with means to connect and extend tools already usedto understand ‘chaos’ in quantum mechanics.

Recent work [13] has already opened an exciting pos-sibility to study the quantum-classical correspondence inintegrable and chaotic systems using DL. In particular, ithas been suggested that a neural network (NN) can learnthe difference between wave functions that correspond tointegrable and chaotic systems. It is important to pur-sue this research direction further, and to understandand interpret how a NN distinguishes the two situations.This information can be used in the future to formulatenew conjectures on the role of classical chaos in quantummechanics. The main challenge here is the extractionof this information from a NN, which often resembles ablack box. Ongoing research on interpretability of NNssuggests certain routes to understand the black box [14–16] (see also recent works that discuss this question forapplications in physical sciences [17–19]). However, thereis no standard approach to this problem. In part, this isconnected to the fact that DL relies on general-purposelearning procedures, therefore, one does not expect that

arX

iv:2

102.

0496

1v2

[qu

ant-

ph]

2 A

ug 2

021

2

there can be a unique way to analyze a neural networkat hand. For example, as we will see, the training of anetwork for the ‘integrable’ vs ‘chaotic’ state recognitionis very similar to the classic dog-or-cat classification.1 Itis not clear, however, that the tools that can be used tointerpret the latter (e.g., based on stable spatial relation-ships [20]) are also useful for the former. In particular, atraining set for the ‘integrable’-or-‘chaotic’ problem con-tains information about vastly different length scales (de-termined by the energy), whereas a training set for catsvs dogs has only length scales given by the size of the an-imal. Therefore, it is imperative to study interpretabilityof neural networks used in physics separately from thatin other applications.

In this paper we analyze a neural network, which hasbeen trained using highly excited states of a triangularbilliard, and attempt to extract the learned features. Bil-liards are conceptually simple systems, yet it is expectedthat they contain all necessary ingredients for study-ing the role of chaos in quantum mechanics [8]. Fur-thermore, eigenstates of quantum billiards are equiva-lent to the eigenstates of the Helmholtz equation withthe corresponding boundary conditions, which connectsquantum billiards and the wave chaos in microwave res-onators [8, 21]. The triangular billiard is one of the most-studied models in quantum chaology [22–26], and there-fore it is well-suited for our study focused on analyzingneural networks as a possible tool for quantum chaology.

In our analysis, we rely on convolutional neutral net-works (ConvNets) for image classification [27], whichhave recently been successfully applied to categorize nu-merical and experimental data in physical sciences [13,28–35]. These advances motivate us to apply ConvNetsto categorize quantum states as integrable and non-integrable. Our goal can be stated as follows: given aset of highly excited states, build a network that canclassify any input state as integrable or not, and, more-over, study features of this network. One comment is inorder here. There are various definitions of quantum in-tegrability [36], so we need to be more specific. In thiswork, we call a quantum system integrable, if it is Bethe-ansatz integrable, i.e., if one can write any eigenstate asa finite superposition of plane waves. We shall also some-times use the word chaotic instead of non-integrable. Fi-nally, we note that the properties ‘integrable’ and ‘non-integrable’ are usually attached to a given physical sys-tem, e.g., following an analysis of global properties likethe distribution of energy levels. However, the corre-spondence principle implies that these labels can also beapplied to individual states of a quantum system. In thispaper, we use both notions and show that they are com-patible. We employ neural networks to analyze the wavefunctions of individual quantum states.

1 The dog-or-cat classifier in this case is a network with one outputlabel for a dog and one for a cat, which has been trained using aset of a few thousand pictures.

We show that a trained network accurately classifiesa state as being ‘integrable’ or ‘non-integrable’, whichimplies that a ConvNet learns certain universal featuresof highly-excited states. We argue that a trained neuralnetwork considers almost any random state generated bya Gaussian, Laplace or other distribution as ‘chaotic’,as long as the state includes a sufficient amount of zerovalues. This observation agrees with our intuition thata non-integrable state has only weak correlations. Wediscuss the effect of the noise and coarse graining in ourclassification scheme, which sets limitations on the appli-cability of neural network to analyze experimental andnumerical data.

Our motivation for this work is thus threefold: First,we want to demonstrate that neural networks can clas-sify the morphology of the three-body states correctly.Therefore, we investigate a known model system withtwo identical fermions and an impurity as a function ofthe impurity mass in order to be able to verify the neu-ral network analysis. Second, we want to analyse theclassifying network and understand the way it operates.Our third goal is to use the network analysis to clarifythe situation regarding suggested new integrable cases forother parameter values than the established ones [37, 38].However, we do not find any evidence of such cases.

The paper is organized as follows. In Sec. II we intro-duce the system at hand: a triangular quantum billiardthat is isomorphic to three impenetrable particles on aring. Its properties are discussed in Sec. III using stan-dard methods. In Sec. IV, we present our neural networkapproach and use it in Sec. V to classify the states ofthe system. Moreover, we analyze the properties of thenetwork. In Sec. VI, we conclude. Some technical detailsare presented in the appendix.

II. FORMULATION

We study billiards isomorphic to the relative motion ofthree impenetrable particles in a ring: two fermions andone impurity. Characteristics of these triangular billiardsare presented below, see also Ref. [39, 40]. Our choiceprovides us with a simple parametrization of triangles interms of the mass ratio, κ = mI/m, where mI (m) is themass of the impurity (fermions). Furthermore, it allowsus to shed light on the problem of three particles in aring with broken integrability [41–43].

For simplicity, we always assume that the impurity isheavier than (or as heavy as) the fermions, correspondingto 1/κ ∈ [0, 1]. As we show below, this leads to a family ofisosceles triangles with the limiting cases (90◦, 45◦, 45◦)for 1/κ = 0 and (60◦, 60◦, 60◦) for κ = 1. These limitingtriangles correspond to two identical hard-core particlesin a square well and a 2+1 Gaudin-Yang model on aring [44], respectively. Both limits are Bethe-ansatz inte-grable, see Refs. [23, 45] for a more detailed discussion.Note that certain extensions to the Bethe ansatz suggestthat additional solvable cases exist [37, 38]. However, our

3

numerical analysis does not find any traces of solvabilitybeyond the two limiting cases, and supports the widelyaccepted idea that almost any one-dimensional problemwith mass imbalance is non-integrable (notable excep-tions are discussed in Refs. [46–50]). Therefore, in thiswork we refer to systems with 1/κ = 0 and 1 as inte-grable, in the sense that they can be analytically solvedusing the Bethe ansatz (cf. Ref. [36]). Systems with othermass ratios are called non-integrable.

A. Hamiltonian

The Hamiltonian of a three-particle system with zero-range interactions reads as

H = − ~2

2m

∂2

∂x21− ~2

2m

∂2

∂x22− ~2

2κm

∂2

∂y2+g∑i

δ(xi−y). (1)

Everywhere below we focus on the limit g → ∞. InEq. (1), 0 < xi < L (0 < y < L) is the coordi-nate of the ith fermion (impurity), while L is the lengthof the ring, see Fig. 1 (a). The eigenstates (φ) of Hare periodic functions in each variable. They are an-tisymmetric with respect to the exchange of fermions,i.e., φ(x1, x2, y) = −φ(x2, x1, y). Furthermore, the limitg → ∞ demands that φ vanishes when the fermion ap-proaches the impurity, i.e., φ(xi → y) → 0. For conve-nience, we use the system of units in which ~ = 1 andm = 1 in the following. For our numerical analysis, wechoose units such that L = π.

The Hamiltonian H can be written as a sum of the rel-ative and center-of-mass parts. To show this, we expandφ using a basis of non-interacting states, i.e.,

φ(x1, x2, y) =∑

n1,n2,n3

a(n3)n1,n2

e−2πiL (n1x1+n2x2+n3y), (2)

where a(n3)n1,n2 = −a(n3)

n2,n1 to satisfy antisymmetric condi-tion on the wave function. It is straightforward to seethat the Hamiltonian does not couple states with differ-ent values of the “total momentum”, P = 2π ntot

L ;ntot =n1 + n2 + n3 because of translational invariance. For ex-ample, the operator δ(x1 − y) couples two states via thematrix element:∫

dx1dx2dyδ(x1−y)e−2πiL (n1−n′

1)x1+(n2−n′2)x2+(n3−n′

3)y),

(3)which equals δn2,n′

2δn1+n3,n′

1+n′3, and, hence, con-

serves P . The integral of motion, P , allows us to writethe wave function as

φ = e−iPy∑n1,n2

a(ntot−n1−n2)n1,n2

e−2πiL (n1(x1−y)+n2(x2−y)),

(4)and define the function, which depends only on the rela-tive coordinates:

ψP (z1, z2) = eiPyφ(x1, x2, y), (5)

where zi = Lθ(y − xi) + xi − y, with the Heaviside stepfunction: θ(x > 0) = 1, θ(x < 0) = 0. The coordinates ziare chosen such that the function ψP (z1, z2) takes valueson zi ∈ [0, L], see Fig. 1 (b).

The function ψP is an eigenstate of the Hamiltonian

HP = −1

2

2∑i=1

∂2

∂z2i− 1

2κ

(2∑i=1

∂

∂zi

)2

+ iP

κ

2∑i=1

∂

∂zi, (6)

which will be the cornerstone of our analysis. As weshow below, it is enough to consider only HP=0 forour purposes. To diagonalize H0, we resort to ex-act diagonalization in a suitable basis. As a basis ele-ment, we use the real functions sin

(n1πz1L

)sin(n2πz2L

)−

sin(n1πz2L

)sin(n2πz1L

), where n1 and n2 are integers with

nmax > n1 > n2 > 0, which is a standard choice for thistype of problems, see, e.g., [51, 52]. Our choice of thebasis ensures that ψP=0 is real for the ground and allexcited states. The parameter nmax defines the maxi-mum element beyond which the basis is truncated. Notethat the basis element is the eigenstate of the systemfor 1/κ = 0. Therefore, we expect exact diagonalizationto perform best for large values of κ and more poorlyfor κ = 1. To estimate the accuracy of our results, webenchmark against the exact solution for an equilateraltriangle (κ = 1), see the discussion in the Appendix.Using nmax = 130, we calculate about 4000 states whoseenergies have relative accuracy of the order of 10−3. Thisset of 4000 states is an input for our analysis in the nextsection.

To summarize this subsection: We perform the trans-formation from H,φ to HP , ψP to eliminate the coordi-nate of the impurity from the consideration. Our pro-cedure can be considered as the Lee-Low-Pines transfor-mation [53] in coordinate space, which is a known toolfor studying many-body systems with impurities in aring [54–57]. Below we argue that HP can be furthermapped onto a triangular billiard. Note however that weare going to work with HP everywhere. Its eigenfunc-tions are defined on a square (see Fig. 1 (b)), allowing usto use them directly as an input for ConvNets.

B. Mapping onto a triangular billiard

It is known that three particles in a ring can be mappedonto a triangular billiard [39, 40]. Here we show thismapping starting with HP . First of all we rotate thesystem of coordinates to eliminate the mixed derivative∂∂z1

∂∂z2

; see Fig. 1 (c). To this end, we introduce the

system of coordinates z = (z2 − z1)/√

2 and Z = (z2 +

z1)/√

2, where the Hamiltonian reads as

HP (z, Z) = −1

2

∂2

∂z2− 1

2

∂2

∂Z2− 1

κ

∂2

∂Z2+ i

√2P

κ

∂

∂Z. (7)

The last term here can be eliminated by a gauge trans-

formation ψP → exp(i√2P

κ+2 Z)ψP . Therefore, in what

4

x1 x2

yz1

z2

Z

z

Z

z

a b

cd

FIG. 1. The figure illustrates the system of interest and thecorrespondence between three particles in a ring and a tri-angular billiard. Panel (a): Three particles in a ring. Twofermions have coordinates x1 and x2, the coordinate of theimpurity is y, see Eq. (1). Panel (b): The coordinates z1 andz2 describe the relative motion of three particles. Panel (c):The coordinates z and Z, which are obtained after rotationof z1 and z2, see Sec. II B). Panel (d): The triangular bil-liard is obtained upon rescaling the coordinates z and Z. Toillustrate the transformation from (a) to (d), we sketch the(real-valued) ground-state wave function for κ = 1 in pan-els (b)-(d). The blue (red) color denotes negative (positive)values of ψP=0 (see. Eq. (5)), which is chosen to be real inour analysis. The intensity matches the absolute value. Notethat in panel (d), we show only z > 0. The wave function forz < 0 is obtained from the fermionic symmetry by reflectionon the Z axis.

follows we only consider P = 0 without loss of generality.We shall omit the subscript, i.e., we write ψ. Note that itis enough to study only z ≥ 0, because of the symmetryof the problem.

To derive the standard Hamiltonian for quantum bil-liards:

h = −1

2

∂2

∂z2− 1

2

∂2

∂Z2, (8)

we rescale and shift the coordinates as z = z, and Z =√κ/(κ+ 2)(Z−L/

√2), see Fig. 1 (d). The Hamiltonian

h is defined on an isosceles triangle with the base angleobtained from tan(α) =

√(κ+ 2)/κ. For systems with

more particles the corresponding transformations H →HP → h lead to quantum billiards in polytopes, allowingone to connect an N -body quantum mechanical problemto a quantum billiard in N − 1 dimensions. This can bea route for finding new applications of solvable models,see Refs. [39, 46, 48].

Finally, we note that if the interaction term in Eq. (1)was an impenetrable wall with some radius R instead ofthe delta function, then the considerations above would

also lead to a mapping of the system onto a triangle.(See Ref. [58] for an illustration with an equilateral tri-angle.)

III. PROPERTIES OF THE SYSTEM

A discussion of highly excited states of triangular bil-liards can be found in the literature [22–26]. However, wefind it necessary to review some known results and calcu-late some new quantities in order to explain our currentunderstanding of the difference between integrable andnon-integrable states. In principle, highly excited statesof a quantum system can be simulated using microwaveresonators (see, e.g., [59, 60]), or generated by means ofFloquet engineering – by choosing the driving frequencyto match the energy difference between the initial andthe desired final state (see, e.g., Ref. [61]). Thereforethe results of this section are not of purely theoreticalinterest, as they can be observed in a laboratory.

As we outlined in the introduction, there are two mainapproaches for analyzing a connection between highly-excited states and classical integrability. The first onerelies on statistical properties of the energy spectra, whilethe second one focuses on the morphology of individualquantum states. This section sets the stage for our fur-ther study by discussing these approaches in more detail.

A. Energy

We start by calculating the energy spectrum. It pro-vides a basic understanding of the evolution from an ‘in-tegrable’ to a ‘chaotic’ system in our work as a func-tion of κ. We present the first 30 states of H0 in Fig. 2(top). Note that an isosceles triangle has a symmetry axis

(Z → −Z), which corresponds to a mirror transforma-tion (in the particle picture this symmetry correspondsto zi → L− zi). The wave function can be symmetric orantisymmetric with respect to the mirror transformationand we consider these cases separately. The former statesare denoted as having p = 1, and the latter have p = −1.

The energy spectrum features inflation of the spacingbetween levels, which can be understood as a repulsion oflevels in the Pechukas gas [62, 63]. According to Weyl’slaw, it can also be interpreted as a change of the densityof states, ρ(κ)(E) = dN/dE, where N(E) is the num-ber of states with the energy less than E. The functionρ(κ)(E) can be easily calculated using Weyl’s law [64] forthe triangular billiard described by the Hamiltonian h:

ρ(κ)(E →∞)→ L2

4π

√κ

κ+ 2. (9)

The density of states is independent of the energy in thisequation because we work with a two-dimensional ob-ject. Equation (9) is derived assuming large values ofE, however, in practice, it also describes well the den-sity of states in a lower part of spectrum (cf. Refs. [24]).

5

0

250

500

750

1000

ener

gy, EL

2

p= − 1

p= + 1

2.5 5.0 7.5 10.0 12.5 15.0mass ratio,

0

20

40

60

80

ener

gy, Eρ

()

FIG. 2. The spectrum of the Hamiltonian HP from Eq. (7) asa function of the mass ratio, κ. The figure presents the first 30states: 15 mirror-symmetric states, p = 1, (red, solid curves),and 15 mirror-antisymmetric states, p = −1, (blue, dashedcurves). The top panel shows the eigenstates of Eq. (6). The

bottom panel shows these energies multipled by ρ(κ) fromEq. (9).

If we multiply the energies presented in Fig. 2 (top) byρ(κ) then we obtain a spectrum without inflation, i.e., alllevels are equally spaced on average, see Fig. 2 (bottom).

Multiplication of E by ρ(κ) is a simple example ofunfolding, which allows us to directly compare featuresof the energy spectrum for different values of κ. Thegoal of the unfolding is to extract the “average” prop-erties of the levels distribution and, thus, diminish theeffect of local level density fluctuations in the spectrum.While there are many possible ways to implement theunfolding procedure, which depend on the properties ofthe energy spectrum (for further information see, e.g.,Refs. [9, 65, 66]), the ultimate goal is to obtain rescaledlevels with unit mean spacing. Below, we rescale all ofthe energy levels by the mean distance between them,thus, obtaining the unit mean spacing. We benchmarkedresults of this unfolding against more complicated ap-proaches, and found qualitatively equivalent outcomes.

We use unfolded spectra to analyze the distributionof nearest neighbors, P (s), which shows the probabilitythat the distances between a random pair of two neigh-boring energy levels is s. The function P (s) is presentedin Fig. 3, see also [24, 25, 45], where some limiting casesare analyzed. For the sake of discussion, we only studythe states with p = 1, however, we have checked that thecase with p = −1 leads to qualitatively identical results.The size of bins in the histograms in Fig. 3 is virtuallyarbitrary [66]. For visual convenience, we have followeda rule of thumb that the number of bins should be takenat approximately a square root of the number of the con-sidered levels.

Figure 3 shows a transition from regular to chaoticwhen the mass ratio changes from κ = 1 to larger val-

0.0

0.5

1.0

P(s

) κ = 1.0 κ = 1.2

0.0

0.5

1.0

P(s

) κ = 2.0 κ = 3.0

0 1 2 3

s

0.0

0.5

1.0

P(s

) κ = 5.0

0 1 2 3

s

1/κ = 0

FIG. 3. The histogram shows the nearest-neighbor distribu-tion, P (s), as a function of s for different values of the massratio, κ. The (red) solid curve shows the Wigner distributionfrom Eq. (10). The (black) dashed curve shows the Poissondistribution, e−s. To produce these figures, only states withp = 1 are used.

ues. The degeneracies in the energy spectrum for κ = 1and 1/κ = 0 lead to well spaced bins in the figure. Thisbehavior is however rather unique, and it is immediatelybroken for other mass ratios. For example, already forκ = 1.2 the levels start to repel each other, and the dis-tribution P (s) can be approximated by the Wigner dis-tribution [8]

PGOE(s) =πs

2e−

πs2

4 . (10)

Note that it is important to use only one value of p forthis conclusion. Levels that correspond to different valuesof p do not repel each other, and the Wigner distributioncannot be realized [24].

It is impossible to analyze every value of κ. However,we can also say something on average about our system.To that end, we calculate the dependence of the minimaldistance between levels as a function of the number ofconsidered levels, δmin(N) = inf{En − En−1}n<N , hereEn is the energy of the nth state with a given value ofp. For a random matrix, δmin is expected to scale as1/√N [67, 68]. To the best of our knowledge, this re-

sult is not strictly proven, but at the intuitive level itcan be understood from Eq. (10). The probability thatthe distance between two energy levels is smaller than

δmin is given by∫ δmin0

PGOEds = 1 − e−πδ2min/4. In thelimit δmin → 0, this expression can be approximatedby πδ2min/4. If we consider N lowest states, then theprobability that all nearest neighbors are separated byδ > δmin is given by (1− πδ2min/4)N . To keep this prob-ability independent of N , the parameter δmin must beproportional to 1/

√N .

We show δmin for our system for κ = 5 in Fig. 4 (leftpanels). We see that for a given value of κ it is im-

6

0.05

0.10

0.15

0.20

0.25

0.30δ m

in

p = +1κ = 5.0

p = +1Average

500 1000 1500 2000

N

0.05

0.10

0.15

0.20

0.25

0.30

δ min

p = −1κ = 5.0

500 1000 1500 2000

N

p = −1Average

FIG. 4. The minimal distance between energy levels as afunction of the number of considered levels. The data pointsare given as (red) crosses. The (black) solid curve is addedto guide the eye. The left panels show δmin for κ = 5 andp = ±1. The right panels display δmin averaged over differentmass ratios κ (see the text for details). The (green) dashedcurves in the right panels show the best-fit to the asymptotic1/√N behavior, which is expected for random matrices.

possible to verify 1/√N scaling, at least for the con-

sidered amount of eigenstates. However, the random-ness present in a mass-imbalanced system can be re-covered. To show this, we average δmin over differentmasses, i.e., δaveragemin = 1

M∑δmin(κi), where M deter-

mines how many values of κ appear in the sum. To pro-duce Fig. 4 (right panels), we sum over the followingvalues of the mass ratios: κ = 1.1, 1.2, ..., 5. The param-eter δaveragemin has approximately 1/

√N behavior at large

values of N , which confirms our expectation that systemswith 1/κ ∈ (0, 1) are not integrable.

B. Wave function

The analysis above shows a drastic change of proper-ties of the system when moving from integrable to non-integrable regimes. Information about this transition isextracted by analyzing the energy levels as in Fig. 3, al-though, the correspondence principle conjectures proper-ties at the level of individual wave functions. The wavefunction of a highly excited state contains too much in-formation for the human eye, and one has to rely on afew existing conjectures that allow one to connect clas-sical chaos to quantum states. For example, the chaoticstates are expected to be similar to a random superpo-sition of plane waves [6], since the underlying classicalphase space has no structure, i.e., the classical motion isnot associated with motion on a two-dimensional torus.This expectation applies to a typical random state (notto atypical, e.g., scared states [69]). In contrast, the wave

0.00

0.80

1.60

2.40

3.20= 1.0 = 1.2

0.00

0.80

1.60

2.40

3.20= 2.0 = 3.0

0.0 0.4 0.8 1.20.00

0.80

1.60

2.40

3.20= 5.0

0.0 0.4 0.8 1.2

1/ = 0

0.0 0.4 0.8|ψ|

0

4

P(|ψ|)

0.0 0.4 0.8|ψ|

0

4

P(|ψ|)

0.0 0.4 0.8|ψ|

0

4

P(|ψ|)

0.0 0.4 0.8|ψ|

0

4

P(|ψ|)

0.0 0.4 0.8|ψ|

0

4

P(|ψ|)

0.0 0.4 0.8|ψ|

0

4

P(|ψ|)

|ψ|, state no. 500P(|ψ|)

FIG. 5. The distributions of values of |ψ| for different massratios, κ = mI/m. The histogram describes the probabilitythat a pixel has a value of |ψ| in a specific interval. For themain plots, we use a 315× 315 pixel raster image of |ψ|. Theinsets are showing P(ψ) for the state represented by 33× 33pixel image. The (red) solid curve shows the prediction ofEq. (11) (no fit parameters).

functions of integrable states are expected to have somenon-trivial morphology, since classical phase space of in-tegrable systems has some structure. Below, we illustratethese ideas for our problem. We focus on a distributionof wave-function amplitudes, although, other signaturesof ‘chaos’ in eigenstates connected to local currents andnodal lines2 [70–72] will also be important when we ana-lyze our neural network.

A celebrated result of the random-wave conjecture isa Gaussian distribution of wave-function amplitudes, seeexamples in Refs. [73–75]:

P(ψ) =1√2πv

e−|ψ|2

2v2 , (11)

where the variance v = 1/L fixes the normalization of thewave function3. We present our numerical calculations

2 A nodal line is a set of points {X,Y } that satisfy ψ(X,Y ) = 0.3 The value of v is calculated by using the average value of

ψ2, i.e., ψ2

=∫x2P (x)dx, in the normalization condition,

i.e.,∫ψ2dz1dz2 = ψ

2 ∫dz1dz2 = 1, which leads to the condi-

tion on v as∫x2P (x)dx = 1/

∫dz1dz2.

7

of P(ψ) in Fig. 5. For this figure, we discretize the wavefunction for the 500th state using either a 315×315 pixelgrid or a 33 × 33 pixel grid, and assign to each unit ofthe grid a value that corresponds to ψ in the center ofthe unit. The distribution of these central values for agiven value of κ is presented as a histogram in Fig. 5.For κ = 1 the P(ψ) resembles an exponential function(cf. Ref. [75]). For larger values of κ, a Gaussian profileis developed. The distinction between the histogram andEq. (11) is clear for κ = 1. For 1/κ = 0 the difference isless evident. Note that the peak at ψ = 0 is enhanced incomparison to the prediction of Eq. (11) for all values ofκ. This is due to the evanescence of the wave functionat the boundaries, which is a finite-size effect beyondEq. (11). Finally, the characteristics of the states arealso visible in a low-resolution images, see the insets ofFig. 5. This feature will be used in the design of ourneural network discussed below.

IV. NEURAL NETWORK

To construct a neural network that can distinguish in-tegrable states from non-integrable ones, we need to

A. prepare a data set for training the network

B. choose a suitable architecture and a training algo-rithm

In this section, we discuss these two items in detail.

A. A data set

As data set we use the set A made of two-dimensionalimages that represent highly excited states. We can useimages of (real) wave functions, ψ(z1, z2) or probabilitydensities, |ψ(z1, z2)|2. We have checked that these tworepresentations lead to similar results. In the paper, wepresent only our findings for |ψ(z1, z2)|2. To produceA, we diagonalize the Hamiltonian HP=0 of Eq. (6) forκ = 1, 2, 5 and 1/κ = 0. Each image has a label – in-tegrable (for κ = 1 and 1/κ = 0) or non-integrable (forκ = 2 and 5).4 We do not include information about themirror symmetry, i.e., states with different values of pare treated on the same footing, since we do not expectthat this information is relevant for a coarse-grained (seebelow) image of |ψ(z1, z2)|2. This allows us to work withtwice as large datasets compared to Fig. (3). Each massratio contributes 1000 states to A, which therefore con-tains 4000 images in total. It is reasonable to not use

4 To avoid any bias towards non-integrable states, we use non-integrable states for only two values of κ. However, we havechecked that our conclusions hold also true if we include othervalues of κ into the data set, in particular, if we add 1000 statesto A from a system with κ = 15.

320× 320

33× 33

64× 64

aliasingoccurs

structure isstill visible

FIG. 6. Schematic representations of a probability distribu-tion of a state for 1/κ = 0 susceptible to aliasing. The leftimage shows a high-resolution representation (320×320 pixelimage). This representation contains too many pixels for ourpurposes, and can be optimized. The right images presenttwo low-resolution representations, which we use to train thenetwork. The representation with 33 × 33 pixels does notcontain enough information and spatial aliasing occurs. Therepresentation with 64× 64 pixels contains all relevant infor-mation, and is used for the analysis in Sec. V.

data sets that contain states with very different energies:Very different energies lead to very different length scales,and hence different information content that should belearned. We choose to include all states from the 50thto 1050th excited states. Not much should be deducibleabout the low-lying states (with N ∼ 10) from the cor-respondence principle, therefore, we do not use them inour study.

A wave function ψ(z1, z2) is a continuous function ofthe variables z1 and z2, see Fig. 1 (b). To use it as an in-put for a network, we need to discretize and coarse-grainit. To this end, we represent ψ as a 64× 64 pixel image,and as the value of the pixel we use the value of the wavefunction at the center of the pixel.5 The resolution is im-portant for this discretization. Low resolution might notbe able to capture oscillations present in highly excitedstates, leading to a loss of important physical informa-tion. For example, the approximately Nth state in thespectrum for 1/κ = 0 will have about

√N oscillations

in each direction, and it is important therefore to use a2√N×2

√N representation of the wave function (similar

to the Nyquist–Shannon sampling theorem)6. For a lower

5 The color depth (i.e., how many colors are available) of a pixel iseffectively given by the numerical precision used to produce theinput data. If experimental data is used as an input, then theiraccuracy will determine the color depth of a pixel.

6 It is worth noting that a general (in particular, non-integrable)

8

resolution, the oscillations are not faithfully reproducedin the low resolution image and spatial aliasing occurs.We illustrate this using the 33×33 resolution in Fig. 6 foran integrable state that is susceptible to spatial aliasing.

Note that out of curiosity, we have also used imageswith 33 × 33 pixel resolution to train our network. Thenetwork could reach relatively high accuracy (higher than90%). However, not all integrable states were detectedproperly. For example, the one in Fig. 6 was classifiedas non-integrable by the network. In general, spatialaliasing is more damaging for integrable states, whichhave symmetries that should be respected; non-integrablestates are more random, and some noise does not changethe classification of the network. Everywhere below weuse the 64 × 64 pixel representation, which gives a suf-ficiently accurate representation of the state, so that wedo not need to worry that the network learns unphys-ical properties. Note that certain local features (e.g.,ψ(z1 = z2) = 0) of the wave function may disappear atthis resolution. The overall high accuracy of our net-work suggests that such features are not important forour analysis.

The set A seems somewhat small. For example, thewell-known Asirra dataset [76] for the cat-dog classifica-tion contains 25000 images that are commonly used totest and compare different image recognition techniques.However, we will see that A is large enough to train anetwork that can accurately classify integrable and non-integrable states. The dataset A is further divided intotwo parts. We draw randomly 85 % of all states and usethem as a training set. The remaining 15 % is used fortesting. We fix the random seed used for drawing to avoiddiscrepancies between different realizations of the net-work. It is worthwhile noting that in image-recognitionapplications, the dataset A may be divided into morethan two parts. For example, in addition to the trainingset and testing set, one can introduce a validation (ordevelopment) set [77], which is used to fine-tune parame-ters of the model. We do not use this additional set here.The focus of this work is on understanding features ofour general image classificator, and not on improving itsaccuracy.

B. Architecture

The neural network in our problem is a map that actsin the space X made of all 64× 64 pixel representationsof |ψ|2. By analogy to the standard dog-vs-cat classifier,the output of the network is a vector with two elementsb. Note that n output neurons are usual for classifying

n classes. However, it is possible to use a single outputneuron for a binary classification, since we know thatb1 + b2 must be equal to one. We use two neurons, sincesuch an architecture can be straightforwardly extendedto more output neurons, which may be useful for thefuture studies, as we discuss in Sec. VI.

The first element of the output layer, 0 ≤ b1 ≤ 1, deter-mines the probability that the input state is integrable,whereas the second element b2 = 1 − b1 is the proba-bility that the input state is non-integrable. An inputstate is classified as integrable (non-integrable) if b1 > b2(b1 < b2).

Mathematically, the network is a map f , which acts onthe element a of X as

f(a; θ, θhyp) = b. (12)

The f is determined by the set of parameters θ, whichare optimized by training. Since our problem is similarto image recognition (in particular dog-vs-cat classifica-tion) [10], which is one of the standard applications inmachine learning, we can use the already known train-ing routines (SGD, ADAM, Adadelta,...) for optimizingθ. The outcome for the parameters θ may vary betweendifferent trainings, and we use this variability to checkthe universality of our results. Specifications of f thatare not trained but specified by the user are called hy-perparameters (θhyp). Examples of them include the lossfunction, optimization algorithm, learning rate, networkarchitecture, size of batches that the data is split into fortraining (batch size) and the length of training (epochs).We find hyperparameters by trial-and-error.

The simplest form of a network is called a dense net-work in which all input neurons are connected to all out-put neurons. However in most cases of image detection,this architecture does not lead to accurate results. Thisalso happens in our case. Instead, we resort to a standardarchitecture based on ConvNets for image recognition,see Fig. 7.7 Our network consists of two convolutionallayers and two max-pooling layers. The former use a setof filters and apply them in parts to the image to producea new smaller image. This is somewhat analogous to arenormalization group transformation [78]. A set of im-ages that are produced by a convolutional layer is calledfeature map. Each convolutional layer is followed by amax-pooling layer which reduces the size of an image.The size of max-pooling layers is a hyperparameter. Inour implementation, max-pooling layers take the largestpixel out of groups of 2 by 2.

One could use architectures different from the one pre-sented in Fig. 7. However, we checked that they do notlead to noticeably different results. Therefore, we do notinvestigate this possibility further.

Nth state might have fewer nodal domains [a nodal domain isa domain bounded by a nodal line] [72], and allow for a repre-sentation with a smaller number of pixels. However, a generaldata set may have states with as many as N nodal domains (cf.

Courant’s theorem), which requires a 2√N×2

√N representation

of the wave function.7 This figure is generated by adapting the code from https://

github.com/gwding/draw_convnet .

9

Featuremaps36@14x14

Featuremaps36@29x29

Featuremaps24@31x31

Featuremaps24@62x62

Inputs1@64x64

Convolution3x3 kernel

Max-pooling2x2 kernel

Convolution3x3 kernel

Max-pooling2x2 kernel

Hiddenunits7056

Hiddenunits128

Outputs2

Flatten Fullyconnected

Fullyconnected

FIG. 7. An illustration of the ConvNet used in our analysis. An input layer, 64× 64 image, is followed by a sequence of layers:a convolutional layer, a pooling layer, a convolutional layer, a pooling layer, and two fully connected layers. The last layer isused to produce an output layer, which is made out of two neurons. In this neural network, we use the rectified linear activationfunction (ReLU).

V. NUMERICAL EXPERIMENTS

Following the discussion in Sec. IV, we train and testthe neural network. We observe that a typical accuracyof the trained network (which we refer later to as N ) is∼ 97%.8 This means that about 18 states out of 600used for testing are given the wrong label. Out of these18 states, roughly one half is integrable. We illustratetypical wave functions that are classified correctly andwrongly in Fig. 8. It does not mean that these statesare in anyway special – another implementation (e.g.,another random seed for weights) will lead to other statesthat are given the wrong label. Non-integrable stateswith some structure (e.g., states with scars) in generalconfuse the network and might be classified as integrable.

In general, it is hard to interpret predictions of the neu-ral network. This becomes clear after noticing that someimages can be changed so that a human eye can hardlydetect any variation. At the same time, this change com-pletely modifies the prediction of the network. Such achange can be accomplished especially easily for inte-grable states9, thus, deep learning (DL) confirms our in-tuition that integrable states are a small subset in thespace of all possible states. However, such a situationcan also occur for non-integrable (in particular scared)states. We illustrate this in Fig. 9, which is obtained byslightly modifying states from A using tools of adversar-ial machine learning, see Ref. [79].

8 We use the word ‘typical’ to emphasize that a trained networkdepends on hyperparameters and random seeds. Even for a givenset of hyperparameters, each set of random parameters leads toa slightly different network N . We can tune hyperparameters toreach higher accuracies. We do not discuss this possibility here,since high accuracy is not the main purpose of our study.

9 Tools of adversarial machine learning can make a neural networkclassify an arbitrary integrable state as ‘non-integrable’ using asmall number (∼ 100) of iterations. This is not possible for anarbitrary non-integrable state.

wrong

inte

grab

le

correct uncertain

non

inte

grab

le

FIG. 8. The figure shows |ψ|2 of exemplary integrable (upperrow) and non-integrable states (lower row) together with thecorresponding prediction of the network. The network assignsa wrong (correct) label for the states in the first (second)column. The third column shows states, which are identifiedcorrectly by the network, but with a low confidence level (withabout 60%). In other words, the states in the third columnconfuse the network and lead to b1 ' b2 in Eq. (12).

One simple way to extract features of the network is tolook at feature maps, which should contain informationabout what features are important. For example, thefirst layer might represent edges at particular parts of theimage, the second might detect specific arrangements ofedges, etc. However, we could not extract any meaningfulinformation from this analysis. This is expected: thefeatures of integrable and non-integrable states are moreabstract and not as intuitive as the features of cats anddogs or images of other objects we encounter in everydaylife.

Other approaches to analyze a network rely on esti-mating the effect of removing a single (or a group) ofelements on a model. For large data sets, this can be

10

initial final initial − final-0.10

-0.05

0.00

0.05

0.10

initial final initial − final-0.050

-0.025

0.000

0.025

0.050

FIG. 9. Fooling the network. The first column shows |ψ|2 ofthe initial state, which is correctly identified by the networkas integrable (non-integrable) in the upper (lower) panel. Thesecond column shows a slightly modified image of |ψ|2, whichis wrongly identified by the network. The third column showsthe difference between the first and second columns dividedby the maximum value of the initial state. The color chartcorresponds to the images in the third graph only.

done by introducing influence functions [80, 81]. Here,we work with a small data set, and, therefore, we cancalculate directly the actual effect of leaving states outof training on a given prediction. Our goal is to under-stand correlations between states of different energies. Inour implementation we compare the prediction of f fromEq. (12) for |ψ|2 to a prediction of f−β for the same state.Here f−β is obtained by training a neural network afterleaving out the set β from A. The comparison of the twopredictions (f−f−β) allows us to estimate the importanceof the set β for the classification of a test state ψ.10 Wepresent a typical example in Fig. 10, where one observesno clear energy correlations, which suggests that the net-work learns different energies simultaneously, at least inthe energy interval we choose to work with. This obser-vation is consistent with our discussion below that thenetwork does not learn specific features of non-integrablestates, and only overall randomness, which does not de-pend on a specific energy window. Finally, we note thatour result in Fig. 10 is an example of cross-validation. Itsuggests weak dependence of the output of the networkon a particular state, which is a necessary condition fora good performance of our neural network.

Below, we explore the network N further. To this end,we resort to numerical experiments. We employ N to an-alyze states outside of the set A. First, we study physicalstates, and then non-physical ones.

10 Note that it is important to choose a test state ψ for whichthe network gives an accurate prediction with high confidencelevel, i.e., bi → 1. For other states, an intrinsic randomness ofConvNets can lead to a drastic change in the classification of thenetwork.

0.00

0.01

0.02

0.03

0.04

f 1−f 1,−β

= 1

1/ = 0

= 2

= 5

0 500 1000 1500 2000 2500energy of LOO state

-0.25

0.00

0.25

0.50

0.75

1.00

1.25

f 1−f 1,−β×

100

FIG. 10. A typical outcome of a leave-one-out (LOO) algo-rithm. (Top) The panel shows f1 − f1,−β for β that consistsof a single state as a function of the energy of the state thatwas left out. (Bottom) The panel shows f1− f1,−β for β thatconsists of ten consecutive states as a function of the energyof the first state in β. The test state, ψ, here is for κ = 5, itsenergy is 534.86 (in our units).

A. Classification of physical states outside of A.

As a first application of N , we use it to classify eigen-states of HP=0 not used in the training, i.e., for κ 6= 1, 2, 5and 1/κ 6= 0. These states are non-integrable (cf. Fig. 4),and we observe that N accurately classifies them as suchas long as κ is far enough from κ = 1, see Fig. 11. Thefigure shows that predictions of N are not accurate onlyfor systems with κ = 1 + ε, where ε is a small parameter.These systems are non-integrable, however, the morphol-ogy of their eigenstates is very similar to the integrableones at κ = 1. The network classifies them wrongly be-cause of this. Already for κ ' 1.5, the accuracy of thenetwork is close to one, and it stays high for larger valuesof κ. The region between 1 and 1.5 can be interpretedas a transition of the network classification from inte-grable to chaotic [13]. We do not expect this region tobe universal – it should depend on hyperparameters, andthe states used for training of N . Therefore, we do notinvestigate it further.

To test the network on integrable states, we use wavefunctions of two non-interacting bosons in a box potentialof size L:

ΨB =Nk1,k2L

[sin(k1z1) sin(k2z2) + sin(k2z1) sin(k1z2)] ,

(13)where k1 ≤ k2, and Nk1,k2 is a normalization constant,

Nk1=k2 = 1 and Nk1<k2 =√

2. The order of the states isdetermined by the energy (k21 + k22)/2. The set of func-

11

1 2 3 4 5mass ratio

0.2

0.4

0.6

0.8

1.0net

work

acc

ura

cy

FIG. 11. Accuracy of the network N as a function of themass imbalance κ. The shaded area shows the uncertaintyarea. To obtain it, we train a set of neural networks usingdifferent random seeds. The highest (lowest) accuracy from aset produces an upper (lower) bound of the uncertainty area.The dots are calculated, and the curves are added to guidethe eye. The points κ = 1, 2, 5, which are used for training,are shown with vertical lines. The point κ = 1 correspondsto an integrable system. All other points of the x-axis areexpected to be non-integrable (cf. Fig. 3).

tions {ΨB} is complementary to the 1/κ = 0 case studiedabove for fermions. The bosonic symmetry yields statesthat are orthogonal to the training set, and therefore,the training routine can have no microscopic informationabout the wave function, ΨB. We use 1000 states of thebosonic type (from the 50th to 1050th) as an input forN . We observe that N accurately (accuracy is ' 96%)classifies states ΨB as integrable.

To connect the analysis of ΨB to studies of quantumbilliards, we note that two bosonic impurities in an infi-nite square well can be mapped on a right triangle withtwo impenetrable boundaries. At the third boundary azero Neumann boundary condition should be satisfied –Ψ′B |z1=z2 = 0. The mapping follows from the mappingdiscussed for fermions (see Fig. 1) assuming that the im-purity is infinitely heavy. In particular, Fig. 1 b) showsthe geometry of the problem in this case. Note thatthe bosonic symmetry requires that the derivative of thewave function vanishes on the diagonal of the square inFig. 1 b). The high accuracy of the classification of thebosonic states suggests that a network trained using theDirichlet boundary condition can also be used to classifystates with the Neumann boundary condition. In otherwords, the network is mainly concerned with the ‘bulk’properties of the wave function, the boundary is not im-portant.

B. Classification of non-physical states

The network N can classify any 64 × 64 pixel inputimage, and it is interesting to explore the outcome of the

0.0 0.3 0.6 0.9 1.2 1.5α

0.0

0.2

0.4

0.6

0.8

1.0

accu

racy

total= 1

1/ = 0= 2= 5

FIG. 12. Prediction of the network for the states from Amultiplied by a factor α. The curves show the accuracies forfour values of the mass ratio κ. The average of these fourcurves is shown as a thick solid curve.

network for images that have no direct physical mean-ing. We start by considering non-normalized eigenstatesof HP=0. The normalization coefficient does not changethe physics behind the states. However, since the func-tion f is non-linear, i.e., f(αx) 6= αf(x), input states musthave the same normalization as the states in the train-ing set for a meaningful interpretation of the network.To illustrate this statement, we use states from A multi-plied by a factor, i.e., we use α|ψ|2 instead of |ψ|2. Fig-ure 12 shows the accuracy11 of the network as a functionof α. The maximum accuracy of the network is reached atα = 1, i.e., for the states used for the training. Integrablestates are classified as non-integrable almost everywhereexcept close to α = 1. A different situation happensfor non-integrable states. They are classified correctlyalmost everywhere, and we conclude that they are lesssusceptible to the factor α. The shape of the curves inFig. 12 is not universal, it depends on hyperparameters ofthe network. However, a general conclusion holds – thenormalization is important, and we use normalized inputfunctions in the further analysis. In principle, it is possi-ble to reduce the sensitivity of our neural network to α.To that end, one needs to add states α|ψ|2 to a trainingset (see data augmentation in deep learning [82]). We donot pursue this possibility here, to demonstrate that thesensitivity of integrable states on α is different from thatfor non-integrable states.

As a next step, we add noise to the images fromA, i.e., we build a new data set using wave functionsψ = aσψ(1 + rσ), where rσ is a noise function whosevalues are drawn from the normal distribution with zeromean and the standard deviation σ; aσ is a normalizationfactor, which is determined for each input state depend-

11 Here we still can talk about the accuracy, since the states αψ2

correspond to integrable or non-integrable situations.

12

0.0 0.5 1.0 1.5 2.0σ

0.0

0.2

0.4

0.6

0.8

1.0ac

cura

cy

total

integr.

non-integr.

FIG. 13. Predictions of the network for the states from Awith noise. The (red) dashed curve shows the accuracy forstates which are integrable for σ = 0 (the accuracy here is de-fined as the percentage of the states identified as integrable).The (green) dotted curve shows the accuracy of the networkfor non-integrable states. The (blue) solid curve shows theaverage of the first two curves.

ing on the function rσ. We assume that rσ possessesbasic symmetries of the problem: fermionic and mirror.Functions ψ naturally appear in applications, and there-fore, it is interesting to investigate the resilience of thenetwork to random noise. We use 4000 states of A withnoise to make a relevant statistical statement, see Fig. 13.Small values of σ lead to weak noise and the network cor-rectly classifies almost all input states. However, largervalues of σ lead to confusing input states, and the net-work fails. It actually fails for integrable states wherethe noise destroys correlations. The accuracy for non-integrable states is always high. The resilience of thenetwork to noise suggests it as a tool to analyze exper-imental data (e.g., obtained using microwave billiards).These experiments [21] can produce a large amount ofdata, however, there are limited variety of tools to ana-lyze the simulated states. In particular, neural networkscan be used to identify atypical states, which do not fitthe overall pattern, e.g., scars.

Our choice of ψ to represent a noisy state is not unique.One could, for example, use instead of ψ the functionψ = aσ(ψ + Grσ), where the parameter G determinesthe relative weight of the function rσ, which is definedas above. Note that G cannot be absorbed in rσ, sincefor a given value of σ, the average amplitude of r2σ iswell defined. For G = 0, the function ψ is a physicalstate, whereas for G → ∞, the function ψ is completelyrandom. In contrast to ψ, the function ψ can becomecompletely independent of the function ψ. This happensif the parameter G is large.

For the data set based upon ψ with small values of G,the accuracy of the network is similar to that presentedin Fig. 13. For large values of G, the network is confusedand classifies states in a random manner. This behav-ior should be compared with the data set ψ for which

0.00

3.90

7.80

11.70

15.60 = 1.0 = 1.2

0.00

3.90

7.80

11.70

15.60 = 2.0 = 3.0

0.00 0.20 0.40 0.600.00

3.90

7.80

11.70

15.60 = 5.0

0.00 0.20 0.40 0.60

1/ = 0

0.00 0.15 0.30|ψ|2

0

4

8

12

16

P(|ψ|2

)

0.00 0.15 0.30|ψ|2

0

4

8

12

16

P(|ψ|2

)

0.00 0.15 0.30|ψ|2

0

4

8

12

16

P(|ψ|2

)

0.00 0.15 0.30|ψ|2

0

4

8

12

16

P(|ψ|2

)

0.00 0.15 0.30|ψ|2

0

4

8

12

16

P(|ψ|2

)

0.00 0.15 0.30|ψ|2

0

4

8

12

16

P(|ψ|2

)

|ψ|2, state no. 500

P(|ψ|2

)

FIG. 14. The histogram shows distributions of probability-density amplitudes for different values of the mass ratio, κ.We use a 315 × 315 pixel representation of the probabilitydensity, |ψ|2, for this analysis. The insets give P(|ψ|2) for theprobability density at the lower (33× 33) pixel resolution.

the states are classified as non-integrable when the noiseis large. To understand this difference, note that ψ re-tains information about the nodal lines of the physicalstate. It turns out that it is important for the inputstate to have enough pixels with small (zero) values (notethat the number of zero pixels is very large for |ψ|2, seeFig. 14). Only such states have a direct meaning for thenetwork, all other states confuse the network and do notallow for extraction of any meaningful information. Itis worthwhile noting that the network does not learn thephysical nodal lines. We checked that almost any randomstate with a large number (' 30− 40%) of zero pixels isclassified as non-integrable.

Our discussion above suggests that the network classi-fies all states with nodal lines but without clear spatialcorrelations as non-integrable. In other words, the net-work perceives almost all states as non-integrable, andthere are only a few islands recognized as integrablestates. Such asymmetry of learning is not expected tohappen for labels with equal significance (cat and dog forexample). We speculate that the observed asymmetry isrelated to the fact that non-integrable states that corre-spond to different energies do not correlate in space [83],and, therefore, a neural network cannot learn any signif-icant patterns when trained with those states.

13

All in all, the state classificator validates our expec-tation that non-integrable states are random and abun-dant, whereas integrable states are rather unique andoccur rarely. For example, in our analysis, integrablestates are a set of measure zero in κ, and perturbationin κ or noise change the prediction of N for these states.The asymmetry also suggests that the standard imple-mentation of deep learning (DL) presented here shouldbe modified to reveal the physics behind non-integrablestates. The present network does not distinguish a non-physical random state with a large number of zero val-ues from a physical non-integrable state since it was nottrained for that question. A possible modification is theaddition of an extra label (b3 in Eq. (12)) for non-physicalstates. Since there are many possible non-physical states,one should frame the problem having an experimental ornumerical set-up in mind, where such states have someorigin and interpretation. We leave an investigation ofthis possibility for future studies.

Finally, we note that the conclusion that the networkclassifies almost all random images as non-integrable isgeneral – it does not depend on the values of hyperparam-eters, initial seed or the distribution (Laplace, Gaussian,etc.) that we use for generation of the random states.We check this by performing a number of numerical ex-periments. In particular, this means that the networkdoes not learn P(ψ) from Eq. (11).

To summarize: A trained network can accuratelyclassify integrable and non-integrable states. The net-work can even classify input states from an orthogonal(bosonic) Hilbert space, on which it can have no micro-scopic information. Thus network identifies generic fea-tures of the ‘integrable’ and ‘non-integrable’ states, al-though this might be not that surprising provided thatwe work with low-resolution 64 × 64 pixel images. Thenetwork classifies almost all random images with nodallines as non-integrable, which suggests that useful infor-mation for the network is mostly contained in integrablestates.

VI. SUMMARY AND OUTLOOK

We used convolutional neural networks to analyzestates of a quantum triangular billiard illustrated inFig. 1. We argued that neural networks can correctlyclassify integrable and non-integrable states. The impor-tant features of the states for the network are the normal-ization of the wave functions and a large number of zeroelements, corresponding to the existence of a nodal line.Almost any random image that satisfies these criteriais classified as non-integrable. All in all, the neural net-work supports our expectation that non-integrable statesare resilient to noise as discussed in subsection V B, andhave a ‘random’ structure, unlike integrable states whosestructure can be revealed by considering, for example,nodal lines.

Our results suggest that machine learning tools can

be used to analyze the morphology of wave functionsof highly excited states obtained numerically or exper-imentally, to solve problems like: find exceptional states(e.g., scars or integrable states) in the spectra, investi-gate the transition from chaotic to integrable dynamics,etc. However, further investigations are needed to setthe limits of applicability for deep learning tools. Forexample, our network considers all states without clearcorrelations as non-integrable. This means that it mustbe modified for the analysis of noisy data, where a noisyimage without any physical meaning could be classified asnon-integrable. To circumvent this, one could introduceadditional labels for training the network. For exam-ple, one could consider three classes – ‘integrable’, ‘non-integrable’, and ‘noise’. This classification might allowfor a more precise description of data sets, and may helpone to extract more information about the physics behindthe problem.

We speculate that a network memorizes integrablestates, and all other states are classified as non-integrable, provided that an image has a large numberof vanishing values. This would explain our findings andit would align nicely with the observation that we ob-serve no overfitting even after many epochs of workingthrough the training set. In the future, it will be inter-esting to use other integrable systems to test this idea. Inparticular, one could use non-triangular billiards, or sys-tems without an impenetrable boundary. For example,one can consider a two-dimensional harmonic oscillatorwith cold atoms. At a single body level, the integrabilityin this system can be broken by potential bumps [84, 85]or spin-orbit coupling [86].

In the present work, we focus on data related to thespatial representation of quantum states. However, ourapproach can also be used to analyze other data. For ex-ample, for few-body cold atom systems, correlation func-tions in momentum space are of interest in theory andexperiment (see, e.g., [87]). Therefore, in the future, itwill be interesting to train neural networks using experi-mental/numerical data that correspond to a momentum-space representation of quantum states, and study thecorresponding features of ‘quantum chaos’. Although,in the present work we focused on two-dimensional data(images of probability densities), it might be interest-ing to consider higher-dimensional analogues (e.g., four-body systems) as well. Further applications include theidentification of phase transitions, the representation ofquantum many-body states, and the solution of many-body problems [88]. Thus machine learning techniquesmay not only be used to classify quantum states but alsoto obtain them by solving the corresponding many-bodyproblem.

To extract further information about the map f , onecould investigate its geometry close to its maximum (min-imum) values. For example, in the vicinity of some accu-rately determined integrable state (f1(x0) = 1), we can

14

write

f1(x0 + δx) ' 1 +1

2δxTGδx, (14)

where the position of the maximum x0 is understood asa vector and G is the Hessian matrix. The first deriva-tive of f1 vanishes since the function f1 is analytic andbounded. Eigenstates of the Hessian G provide us withthe most important correlations. A preliminary studyshows that there are only a handful of eigenvalues of Gfor our network, which suggests the next step in the anal-ysis of our image classifier.

A deeper understanding of the way a neural networkworks may be obtained by combining different tech-niques to interpret its operation. The potential of thisapproach was illustrated in Ref. [15], for instance, byanalyzing how a network looking at images of a labradorretriever detects floppy ears and how that influences itsclassification.

ACKNOWLEDGMENTS

We thank Aidan Tracy for his input during the ini-tial stages of this project. We thank Nathan Harsh-

man, Achim Richter, Wojciech Rzadkowski, and DaneHudson Smith for helpful discussions and comments onthe manuscript. This work has been supported by Euro-pean Union’s Horizon 2020 research and innovation pro-gramme under the Marie Sk lodowska-Curie Grant Agree-ment No. 754411 (A.G.V.); by the German Aeronau-tics and Space Administration (DLR) through Grant No.50 WM 1957 (O.V.M.); by the Deutsche Forschungsge-meinschaft through Project VO 2437/1-1 (Projektnum-mer 413495248) (A.G.V. and H.W.H.); by the DeutscheForschungsgemeinschaft through Collaborative ResearchCenter SFB 1245 (Projektnummer 279384907) and bythe Bundesministerium fur Bildung und Forschung un-der contract 05P18RDFN1 (H.W.H). H.W.H. also thanksthe ECT* for hospitality during the workshop ”Universalphysics in Many-Body Quantum Systems – From Atomsto Quarks”. This infrastructure is part of a project thathas received funding from the European Union’s Hori-zon 2020 research and innovation programme under grantagreement No 824093.

APPENDIX

To generate the input for a neural network, we diagonalize the Hamiltonian HP=0 from Eq. (6) in a truncatedHilbert space whose basis element is

ξn1,n2(x1, x2) = N

[sin(n1πx1

L

)sin(n2πx2

L

)− sin

(n1πx2L

)sin(n2πx1

L

)], (15)

where c > n1 > n2 > 0, c is the cutoff parameter, and N is the normalization constant. In this basis, matrix elementsof the Hamiltonian read as (in our numerical analysis, we use units in which L = π)∫ π

0

∫ π

0

ξm1,m2(x1, x2)∗H0ξn1,n2

(x1, x2)dx1dx2 =1

2(n21 + n22)

(1 +

1

κ

)(δm1,n1

δm2,n2− δm1,n2

δm2,n1)

+n1n2π2κ

[I(m1 + n2,m2 + n1) + I(m1 + n2,m2 − n1)

+ I(m1 − n2,m2 + n1) + I(m1 − n2,m2 − n1)

− I(m1 + n1,m2 − n2)− I(m1 + n1,m2 − n2)

−I(m1 − n1,m2 + n2)− I(m1 − n1,m2 − n2)] ,

(16)

where I(s, t) = 1st [(−1)s − 1] [(−1)t − 1] if s, t 6= 0, and I(s, t) = 0 otherwise. To write these matrix elements in a

matrix form, we use the index

n = n2 − n1 + c(n1 − 1)− (n1 − 1)n12

. (17)

The parameters n1 and n2 can be uniquely defined as:

n1 =1 + 2c

2−√c2 − c− 2n+

9

4, n2 = n+ n1 − c(n1 − 1) +

(n1 − 1)n12

. (18)

15

To choose the cutoff parameter, c, we should find a good balance between the calculation time and the accuracy ofour results. To quantify the accuracy, we compute energies for κ = 1 by diagonalizing H0, and compare them to theexact ones obtained with the Bethe ansatz:

EBA =

n21 + n22 + n23,

∑ni = 0,

(n1 + 13 )2 + (n2 + 1

3 )2 + (n3 + 13 )2,

∑ni = −1,

(n1 + 23 )2 + (n2 + 2

3 )2 + (n3 + 23 )2,

∑ni = −2.

(19)

This solution assumes that the total momentum is zero. The relative difference

ε =EBA − E(c)

EBA(20)

provides us a measure for the accuracy. Note that our exact diagonalization method is expected to work better forκ > 1, since ξ is the eigenstate of a system with 1/κ = 0. The input for a neural network is obtained using c = 130,for which we obtain 726 (3795) states with ε < 10−4 (10−3) within a short enough computation time.

[1] M Born, Atomic Physics, 6th edition (New York: HafnerPublishing Co., 1957) p. 103.

[2] Michael Victor Berry, Ian Colin Percival, and Nigel Os-car Weiss, “The bakerian lecture, 1987. quantum chaol-ogy,” Proceedings of the Royal Society of London.A. Mathematical and Physical Sciences 413, 183–198(1987).

[3] Gordon Belot and John Earman, “Chaos out of order:Quantum mechanics, the correspondence principle andchaos,” Studies in History and Philosophy of Science PartB: Studies in History and Philosophy of Modern Physics28, 147 – 182 (1997).

[4] Wojciech Hubert Zurek, “Decoherence, einselection, andthe quantum origins of the classical,” Rev. Mod. Phys.75, 715–775 (2003).

[5] Michael Victor Berry, M. Tabor, and John Michael Zi-man, “Level clustering in the regular spectrum,” Pro-ceedings of the Royal Society of London. A. Mathemati-cal and Physical Sciences 356, 375–394 (1977).

[6] M V Berry, “Regular and irregular semiclassical wave-functions,” Journal of Physics A: Mathematical and Gen-eral 10, 2083–2091 (1977).

[7] O. Bohigas, M. J. Giannoni, and C. Schmit, “Charac-terization of chaotic quantum spectra and universality oflevel fluctuation laws,” Phys. Rev. Lett. 52, 1–4 (1984).

[8] H.-J. Stockmann, Quantum Chaos - An Introduction(University Press, Cambridge, 1999).

[9] F. Haake, Quantum Signatures of Chaos, 2nd edition(Springer, Berlin, 2001).

[10] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,”Nature 521, 436–444 (2015).

[11] Giuseppe Carleo, Ignacio Cirac, Kyle Cranmer, Lau-rent Daudet, Maria Schuld, Naftali Tishby, Leslie Vogt-Maranto, and Lenka Zdeborova, “Machine learningand the physical sciences,” Rev. Mod. Phys. 91, 045002(2019).

[12] Michael Berry, “Quantum chaology, not quantum chaos,”Physica Scripta 40, 335–336 (1989).

[13] Y. A. Kharkov, V. E. Sotskov, A. A. Karazeev, E. O.Kiktenko, and A. K. Fedorov, “Revealing quantum

chaos with machine learning,” Phys. Rev. B 101, 064406(2020).

[14] Riccardo Guidotti, Anna Monreale, Salvatore Ruggieri,Franco Turini, Fosca Giannotti, and Dino Pedreschi,“A survey of methods for explaining black box models,”ACM Comput. Surv. 51 (2018), 10.1145/3236009.

[15] Chris Olah, Arvind Satyanarayan, Ian Johnson, ShanCarter, Ludwig Schubert, Katherine Ye, and AlexanderMordvintsev, “The building blocks of interpretability,”Distill (2018), 10.23915/distill.00010.

[16] Q. Zhang and S. Zhu, “Visual interpretability for deeplearning: a survey,” Frontiers Inf Technol Electronic Eng19, 27–39 (2018).

[17] Bastian Kaspschak and Ulf-G. Meißner, “How ma-chine learning conquers the unitary limit,” (2020),arXiv:2003.09137 [physics.comp-ph].

[18] Bastian Kaspschak and Ulf-G. Meißner, “A neural net-work perturbation theory based on the born series,”(2020), arXiv:2009.03192 [cs.LG].

[19] Anna Dawid, Patrick Huembeli, Michal Tomza, MaciejLewenstein, and Alexandre Dauphin, “Phase detectionwith neural networks: interpreting the black box,” NewJournal of Physics 22, 115001 (2020).

[20] Q. Zhang, X. Wang, R. Cao, Y. N. Wu, F. Shi, andS. Zhu, “Extracting an explanatory graph to interpret acnn,” IEEE Transactions on Pattern Analysis and Ma-chine Intelligence , 1–1 (2020).

[21] A. Richter, “Playing billiards with microwaves — quan-tum manifestations of classical chaos,” in Emerging Ap-plications of Number Theory , edited by Dennis A. Hejhal,Joel Friedman, Martin C. Gutzwiller, and Andrew M.Odlyzko (Springer New York, New York, NY, 1999) pp.479–523.

[22] M. V. Berry and M. Wilkinson, “Diabolical points in thespectra of triangles,” Proceedings of the Royal Society ofLondon. Series A, Mathematical and Physical Sciences392, 15–43 (1984).

[23] Wai-Kee Li and S. M. Blinder, “Solution of theschrodinger equation for a particle in an equilateral tri-angle,” Journal of Mathematical Physics 26, 2784–2786

16

(1985).[24] David L. Kaufman, Ioan Kosztin, and Klaus Schulten,

“Expansion method for stationary states of quantum bil-liards,” American Journal of Physics 67, 133–141 (1999).

[25] F. M. de Aguiar, “Quantum properties of irrational tri-angular billiards,” Phys. Rev. E 77, 036201 (2008).

[26] T. Araujo Lima, S. Rodrıguez-Perez, and F. M.de Aguiar, “Ergodicity and quantum correlations in ir-rational triangular billiards,” Phys. Rev. E 87, 062902(2013).

[27] Waseem Rawat and Zenghui Wang, “Deep convolutionalneural networks for image classification: A comprehen-sive review,” Neural Computation 29, 2352–2449 (2017),pMID: 28599112.

[28] Lei Wang, “Discovering phase transitions with unsuper-vised learning,” Phys. Rev. B 94, 195105 (2016).

[29] P. Broecker, J. Carrasquilla, R.G. Melko, and et al.,“Machine learning quantum phases of matter beyond thefermion sign problem,” Sci Rep 7, 8823 (2017).

[30] Wenjian Hu, Rajiv R. P. Singh, and Richard T.Scalettar, “Discovering phases, phase transitions, andcrossovers through unsupervised machine learning: Acritical examination,” Phys. Rev. E 95, 062122 (2017).

[31] Y. Zhang, A. Mesaros, K. Fujita, and et al., “Machinelearning in electronic-quantum-matter imaging experi-ments,” Nature 570, 484–490 (2019).

[32] B.S. Rem, N. Kaming, M. Tarnowski, and et al., “Iden-tifying quantum phase transitions using artificial neuralnetworks on experimental data.” Nat. Phys. 15, 917–920(2019).

[33] A. Bohrdt, C.S. Chiu, G. Ji, and et al., “Classifyingsnapshots of the doped hubbard model with machinelearning,” Nat. Phys. 15, 921–924 (2019).

[34] J. Pekalski, W. Rzadkowski, and A. Z. Panagiotopou-los, “Shear-induced ordering in systems with competinginteractions: A machine learning study,” The Journal ofChemical Physics 152, 204905 (2020).

[35] W Rzadkowski, N Defenu, S Chiacchiera, A Trombettoni,and G Bighin, “Detecting composite orders in layeredmodels via machine learning,” New Journal of Physics22, 093026 (2020).

[36] Jean-Sebastien Caux and Jorn Mossel, “Remarks on thenotion of quantum integrability,” Journal of Statisti-cal Mechanics: Theory and Experiment 2011, P02023(2011).

[37] You-Quan Li and Zhong-Shui Ma, “Exact results of ahard-core interacting system with a single impurity,”Phys. Rev. B 52, R13071–R13074 (1995).

[38] J.B. McGuire and C. Dirk, “Extending the bethe ansatz:The quantum three-particle ring,” Journal of StatisticalPhysics 102, 971 (2001).

[39] H R Krishnamurthy, H S Mani, and H C Verma, “Ex-act solution of the schrodinger equation for a particle ina tetrahedral box,” Journal of Physics A: Mathematicaland General 15, 2131–2137 (1982).

[40] S.L. Glashow and L. Mittag, “Three rods on a ring andthe triangular billiard,” J Stat Phys 87, 937–941 (1997).

[41] Austen Lamacraft, “Diffractive scattering of three parti-cles in one dimension: A simple result for weak violationsof the yang-baxter equation,” Phys. Rev. A 87, 012707(2013).

[42] Rafael E. Barfknecht, Ioannis Brouzos, and Angela Foer-ster, “Contact and static structure factor for bosonic andfermionic mixtures,” Phys. Rev. A 91, 043640 (2015).

[43] S.K. Joseph and M.A.F. Sanjuan, “Entanglement en-tropy in a triangular billiard,” Entropy 18, 79 (2016).

[44] Xi-Wen Guan, Murray T. Batchelor, and Chaohong Lee,“Fermi gases in one dimension: From bethe ansatz toexperiments,” Rev. Mod. Phys. 85, 1633–1691 (2013).

[45] H.C. Schachner and G.M. Obermair, “Quantum billiardsin the shape of right triangles,” Z. Physik B - CondensedMatter 95, 113–119 (1994).

[46] Maxim Olshanii and Steven G Jackson, “An exactly solv-able quantum four-body problem associated with thesymmetries of an octacube,” New Journal of Physics 17,105005 (2015).

[47] N.J.S. Loft, A.S. Dehkharghani, N.P. Mehta, A. G. Volos-niev, and N. T. Zinner, “A variational approach torepulsively interacting three-fermion systems in a one-dimensional harmonic trap,” Eur. Phys. J. D 69, 65(2015).

[48] T. Scoquart, J. J. Seaward, S. G. Jackson, and M. Ol-shanii, “Exactly solvable quantum few-body systems as-sociated with the symmetries of the three-dimensionaland four-dimensional icosahedra,” SciPost Phys. 1, 005(2016).

[49] N. L. Harshman, Maxim Olshanii, A. S. Dehkharghani,A. G. Volosniev, Steven Glenn Jackson, and N. T. Zin-ner, “Integrable families of hard-core particles with un-equal masses in a one-dimensional harmonic trap,” Phys.Rev. X 7, 041001 (2017).

[50] Yanxia Liu, Fan Qi, Yunbo Zhang, and Shu Chen,“Mass-imbalanced atoms in a hard-wall trap: An exactlysolvable model associated with d6 symmetry,” iScience22, 181 – 194 (2019).

[51] A.G. Miltenburg and T.h.W. Ruijgrok, “Quantum as-pects of triangular billiards,” Physica A: Statistical Me-chanics and its Applications 210, 476 – 488 (1994).

[52] A S Dehkharghani, A G Volosniev, and N T Zinner, “Im-penetrable mass-imbalanced particles in one-dimensionalharmonic traps,” Journal of Physics B: Atomic, Molecu-lar and Optical Physics 49, 085301 (2016).

[53] T. D. Lee, F. E. Low, and D. Pines, “The motion ofslow electrons in a polar crystal,” Phys. Rev. 90, 297–302 (1953).

[54] A. G. Volosniev and H.-W. Hammer, “Analytical ap-proach to the bose-polaron problem in one dimension,”Phys. Rev. A 96, 031601 (2017).

[55] G. Panochko and V. Pastukhov, “Mean-field constructionfor spectrum of one-dimensional bose polaron,” Annals ofPhysics 409, 167933 (2019).

[56] S. I. Mistakidis, A. G. Volosniev, N. T. Zinner, andP. Schmelcher, “Effective approach to impurity dynamicsin one-dimensional trapped bose gases,” Phys. Rev. A100, 013619 (2019).

[57] J. Jager, R. Barnett, M. Will, and M. Fleischhauer,“Strong-coupling bose polarons in one dimension: Con-densate deformation and modified bogoliubov phonons,”Phys. Rev. Research 2, 033142 (2020).

[58] Robert Nyden Hill, “An exactly solvable one-dimensionalthree-body problem with hard cores,” Journal of Mathe-matical Physics 21, 1083–1085 (1980).

[59] H.-J. Stockmann and J. Stein, ““quantum” chaos inbilliards studied by microwave absorption,” Phys. Rev.Lett. 64, 2215–2218 (1990).

[60] S. Sridhar and E. J. Heller, “Physical and numerical ex-periments on the wave mechanics of classically chaoticsystems,” Phys. Rev. A 46, R1728–R1731 (1992).

17

[61] F Lenz, B Liebchen, F K Diakonos, and P Schmelcher,“Resonant population transfer in the time-dependentquantum elliptical billiard,” New Journal of Physics 13,103019 (2011).

[62] Philip Pechukas, “Distribution of energy eigenvalues inthe irregular spectrum,” Phys. Rev. Lett. 51, 943–946(1983).

[63] T. Yukawa, “New approach to the statistical propertiesof energy levels,” Phys. Rev. Lett. 54, 1883–1886 (1985).

[64] V. Ivrii, “100 years of weyl’s law.” Bull. Math. Sci. 6,379–452 (2016).

[65] Oriol Bohigas and Marie-Joya Giannoni, Mathematicaland Computational Methods in Nuclear Physics, editedby J. S. Dehesa, J. M. G. Gomez, and A. Polls (SpringerBerlin Heidelberg, 1984) pp. 1–99.

[66] T. Prosen and M. Robnik, “Energy level statistics in thetransition region between integrability and chaos,” Jour-nal of Physics A: Mathematical and General 26, 2371–2387 (1993).

[67] Gerard Ben Arous and Paul Bourgade, “Extreme gapsbetween eigenvalues of random matrices,” Ann. Probab.41, 2648–2681 (2013).

[68] Valentin Blomer, Jean Bourgain, Maksym Radziwill,and Zeev Rudnick, “Small gaps in the spectrum of therectangular billiard,” Annales Scientifiques de l’EcoleNormale Superieure 50, 1283–1300 (2017).

[69] L. Kaplan and E.J. Heller, “Linear and nonlinear theoryof eigenfunction scars,” Annals of Physics 264, 171 – 206(1998).

[70] John R. Evans and Mark I. Stockman, “Turbulence andspatial correlation of currents in quantum chaos,” Phys.Rev. Lett. 81, 4624–4627 (1998).

[71] K. F. Berggren, K. N. Pichugin, A. F. Sadreev, andA. Starikov, “Signatures of quantum chaos in the nodalpoints and streamlines in electron transport through bil-liards,” Jetp Lett. 70, 403–409 (1999).

[72] Sudhir Ranjan Jain and Rhine Samajdar, “Nodal por-traits of quantum billiards: Domains, lines, and statis-tics,” Rev. Mod. Phys. 89, 045005 (2017).

[73] M. Shapiro and G. Goelman, “Onset of chaos in an iso-lated energy eigenstate,” Phys. Rev. Lett. 53, 1714–1717(1984).

[74] Steven W. McDonald and Allan N. Kaufman, “Wavechaos in the stadium: Statistical properties of short-wavesolutions of the helmholtz equation,” Phys. Rev. A 37,3067–3086 (1988).

[75] Rhine Samajdar and Sudhir R. Jain, “Exact eigenfunc-tion amplitude distributions of integrable quantum bil-liards,” Journal of Mathematical Physics 59, 012103(2018).

[76] J. Elson, J. Douceur, J. Howell, and J.J. Saul, “Asirra:A captcha that exploits interest-aligned manual imagecategorization,” in Proceedings of 14th ACM Conferenceon Computer and Communications Security (CCS) (As-sociation for Computing Machinery, Inc. (ACM), 2007)p. 366.

[77] Gareth James, Daniela Witten, Trevor Hastie, andRobert Tibshirani, An Introduction to Statistical Learn-ing (Springer, New York, NY, 2013).

[78] Kenneth G. Wilson, “The renormalization group andcritical phenomena,” Rev. Mod. Phys. 55, 583–600(1983).

[79] Christian Szegedy, Wojciech Zaremba, Ilya Sutskever,Joan Bruna, Dumitru Erhan, Ian Goodfellow, andRob Fergus, “Intriguing properties of neural networks,”(2014), arXiv:1312.6199 [cs.CV].

[80] Frank R. Hampel, “The influence curve and its role inrobust estimation,” Journal of the American StatisticalAssociation 69, 383–393 (1974).

[81] Pang Wei Koh and Percy Liang, “Understanding black-box predictions via influence functions,” in InternationalConference on Machine Learning, ICML’17, Vol. 70(JMLR.org, 2017) p. 1885–1894.

[82] C. Shorten and T. M. Khoshgoftaar, “A survey on imagedata augmentation for deep learning,” J Big Data 6, 60(2019).

[83] V. N. Prigodin, “Spatial structure of chaotic wave func-tions,” Phys. Rev. Lett. 74, 1566–1569 (1995).

[84] N. L. Harshman, “Infinite barriers and symmetries for afew trapped particles in one dimension,” Phys. Rev. A95, 053616 (2017).

[85] J. Keski-Rahkonen, A. Ruhanen, E. J. Heller, andE. Rasanen, “Quantum lissajous scars,” Phys. Rev. Lett.123, 214101 (2019).

[86] O V Marchukov, A G Volosniev, D V Fedorov, A SJensen, and N T Zinner, “Statistical properties of spec-tra in harmonically trapped spin–orbit coupled systems,”Journal of Physics B: Atomic, Molecular and OpticalPhysics 47, 195303 (2014).

[87] Andrea Bergschneider, Vincent M. Klinkhamer, Jan Hen-drik Becher, Ralf Klemt, Lukas Palm, Gerhard Zurn, Se-lim Jochim, and Philipp M. Preiss, “Experimental char-acterization of two-particle entanglement through posi-tion and momentum correlations,” Nature Physics 15,640–644 (2019).

[88] Sankar Das Sarma, Dong-Ling Deng, and Lu-MingDuan, “Machine learning meets quantum physics,”Physics Today 72, 48 (2019).