Abstract
Diverse manybody systems, from soap bubbles to suspensions to polymers, learn and remember patterns in the drives that push them far from equilibrium. This learning may be leveraged for computation, memory, and engineering. Until now, manybody learning has been detected with thermodynamic properties, such as work absorption and strain. We progress beyond these macroscopic properties first defined for equilibrium contexts: We quantify statistical mechanical learning using representation learning, a machinelearning model in which information squeezes through a bottleneck. By calculating properties of the bottleneck, we measure four facets of manybody systemsâ€™ learning: classification ability, memory capacity, discrimination ability, and novelty detection. Numerical simulations of a classical spin glass illustrate our technique. This toolkit exposes selforganization that eludes detection by thermodynamic measures: Our toolkit more reliably and more precisely detects and quantifies learning by matter while providing a unifying framework for manybody learning.
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Introduction
Manybody systems can learn and remember patterns of drives that propel them far from equilibrium. Such behaviors have been predicted and observed in many settings, from chargedensity waves^{1,2} to nonBrownian suspensions^{3,4,5}, polymer networks^{6}, soapbubble rafts^{7}, and macromolecules^{8}. Such learning holds promise for engineering materials capable of memory and computation. Detecting such learning can also help us understand granular systems, e.g., infer the history of forces experienced by an asteroid core. This potential for applications, with experimental accessibility and ubiquity, have earned these classical nonequilibrium manybody systems much attention recently^{9}.
A classical, randomly interacting spin glass exemplifies driven matter that learns. Let us call a set \(\{ {\vec {A}}, {\vec {B}}, {\vec {C}} \}\) of magnetic fields a drive. Consider randomly selecting a field from the drive and applying it to the spin glass, then repeating this process many times. The spins absorb work from the fields. The power absorbed shrinks adaptively, in a certain parameter regime: The spins migrate toward a corner of configuration space where their configuration approximately withstands the driveâ€™s insults. If new fields are imposed, the absorbed power spikes. If fields from \(\{ {\vec {A}}, {\vec {B}}, {\vec {C}} \}\) are reimposed, the absorbed power spikes again, but less than under the unfamiliar fields^{10}. The spin glass recognizes the original drive.
A simple, lowdimensional property of the materialâ€”absorbed powerâ€”distinguishes drive inputs that fit a pattern from drive inputs that do not. This property reflects a structural change in the spin glassâ€™s configuration. The change is longlived and not easily erased by new stimuli. For these reasons, we say that the material has learned the drive.
Manybody learning has been quantified with properties commonplace in thermodynamics. Examples include power, as explained above, and strain in polymers that learn stress amplitudes. Such thermodynamic diagnoses offer insights but suffer from two shortcomings. First, the thermodynamic properties vary from system to system. For example, work absorption characterizes the spin glassâ€™s learning; strain characterizes nonBrownian suspensionsâ€™. A more general approach would facilitate comparisons and standardize analyses. Second, thermodynamic properties were defined for macroscopic equilibrium states. Such properties do not necessarily describe farfromequilibrium systemsâ€™ learning optimally.
Separately from manybody systemsâ€™ learning, machine learning has flourished over the past decade^{11,12}. Machine learning has helped elucidate how natural and artificial systems learn. Neural networks developed over the past decade can undergo representation learning^{13} (Fig. 1a). Such a neural network receives a highdimensional variable X. Examples include a sentence missing a word, e.g., â€śThe \(\underline{\quad \quad }\) is shining.â€ť The neural network compresses the input into a lowdimensional latent variable Z, e.g., word types and relationships. The neural network decompresses Z into a prediction \({\hat{Y}}\) of a highdimensional variable Y. In the example, Y can be the word missing from the sentence, and \({\hat{Y}}\) can be â€śsun.â€ť The size of the bottleneck Z controls a tradeoff between the memory consumed and the predictionâ€™s accuracy. We call the neural networks that perform representation learning bottleneck neural networks.
In this paper, we construct and deploy a bottleneck neural network to quantify how much manybody systems learn about the patterns of drives that force them: We use representation learning to learn how much manybody systems learn. Our measurement protocols share the following structure (Fig. 2b): The manybody system is trained with a drive (e.g., fields \(\vec {A}\), \(\vec {B}\), and \(\vec {C}\)). Then, the system is tested (e.g., with a field \(\vec {D}\)). Training and testing are repeated in many trials. Configurations realized by the manybody system are used to train a bottleneck neural network via unsupervised learning. Finally, we calculate properties of the neural networkâ€™s bottleneck. We illustrate with numerical simulations of the spin glass, whose learning has been detected with work absorption^{10}. Our methods generalize to other platforms, however. This machinelearning toolkit offers three advantages.

1.
Bottleneck neural networks register learning behaviors more thoroughly and precisely than work absorption.

2.
Our framework encompasses a wide class of strongly driven manybody systems. Although we illustrate with the example of a spin glass, the framework does not rely on any particular thermodynamic property tailored to spins. Our neural network scores a manybody systemâ€™s learning behaviors with dimensionless numbers that can be compared across platforms.

3.
Our approach unites machine learning with learning by manybody systems. The union is conceptually satisfying.
We measure four facets of manybody learning: classification ability, memory capacity, discrimination ability, and novelty detection. Our techniques, however, can be extended to other facets.
Results
First, we introduce our bottleneck neural network. Then, we define the spin glass on which we will test our machinelearning toolkit. We finally show how to quantify, using representation learning, how much a manybody system learns about a drive.
Bottleneck neural network
Representation learning, we argue, shares its structure with a problem in nonequilibrium statistical mechanics (Fig.Â 1b). Consider a manybody system subject to a strong drive. The systemâ€™s microstate occupies a highdimensional space, like the input X to a bottleneck neural network. A macrostate synopsizes the microstate in a few numbers, such as particle number and magnetization. This synopsis parallels the latent space Z. If the manybody system has learned the drive, the macrostate encodes the drive. One may reconstruct the drive from the macrostate, as a bottleneck neural network reconstructs Y from Z. See Ref.^{14} for a formal parallel between representation learning and equilibrium thermodynamics.
We construct a neural network inspired by this parallel. As the macrostate informs computations in the statisticalmechanics problem described above, the neural networkâ€™s bottleneck informs our computations. One might initially aim for a bottleneck neural network that predicts drives from configurations X. But such a neural network would undergo supervised learning, if constructed according to the state of the art of when this paper was written. During supervised learning, the neural network receives tuples (configuration of the manybody system, label of drive that generated the configuration). The drive labels are not directly available to the manybody system. So successful predictions by neural network predictions would not necessarily reflect only learning by the manybody system. Hence we design a bottleneck neural network that performs unsupervised learning, receiving only configurations.
This neural network is a variational autoencoder^{15,16,17}, a generative model: It receives samples x from a distribution over the possible X values, creates a variational model for the distribution, and samples from the model. The model is refined via Bayesian variational inference (see Supplementary NoteÂ I for an overview). The modelâ€™s parameters are optimized via backpropagation during training.
Our variational autoencoder has five fully connected hidden layers, with neuron numbers 200200(number of Z neurons)200200. We usually restrict the latent variable Z to 2â€“4 neurons. This choice enables us to visualize the latent space and suffices to quantify the spin glassâ€™s learning. Growing the manybody system may require more latent dimensions, as may growing the number of drives whose patterns the manybody system must learn. But our studies suggest that the number of dimensions needed \(\ll\) the system size.
FigureÂ 3 depicts the latent space Z. Each neuron corresponds to one axis and represents a continuousvalued real number. The latent space was formed via the protocol detailed below, in the section â€śHow to quantify a manybody systemâ€™s learning of a drive, using representation learning.â€ť To synopsize, we trained the spin glass on one drive in each of 1000 trials; trained the spin glass in another drive in each of 1000 trials; and so on, for five drives total. On the endoftrial spinglass configurations, the neural network was trained. The neural network compressed each configuration to a dot in latent space. We colored each dot according to which drive produced the corresponding configuration. We added the colors after the neural networkâ€™s training, so the neural network received no configurationsâ€™ drive labels. Samecolor dots cluster together, so the spin glass distinguished the drives, as recognized by the neural network.
One might wonder whether our toolkit requires deep learning. Could simpler algorithms detect and measure manybody learning as sensitively? Supplementary NoteÂ II responds negatively. We compare our neural network with simpler competitors that perform unsupervised learning: a singlelayer linear neural network, related to principalcomponent analysis^{18}, and a clustering algorithm. The bottleneck neural network outperforms both competitors. (Competitors that perform supervised learning would enjoy an unfair advantage and, as explained above, would not reflect the manybodyâ€™s system learning faithfully.)
Spin glass
A spin glass exemplifies the manybody learner^{10}. We illustrate our machinelearning toolkit by simulating a glass of \(N= 256\) classical spins. The jth spin occupies one of two possible states: \(s_j = \pm 1\).
The spins couple together and experience an external magnetic field: Spin j evolves under a Hamiltonian
and the spin glass evolves under \(H(t) = {\frac{1}{2}} \sum _{j = 1}^NH_j(t)\), at time t. We call the first term in Eq.Â (1) the interaction energy and the second term the field energy. The couplings \(J_{j k} = J_{kj}\) are defined through an ErdĂ¶sRĂ©nyi random network: Spins j and k have some probability p of interacting, for all j and \(k \ne j\). Each spin couples to eight other spins, on average. The nonzero couplings \(J_{j k}\) are selected according to a normal distribution of standard deviation 1.
\(A_j(t)\) denotes the magnitude and sign of the external field experienced by spin j at time t. The field always points along the same direction (the zaxis), so we omit the arrow from \(\vec {A}_j(t)\). We will simplify the notation for the field from \(\{ A_j(t) \}_j\) to A (or B, etc.). Each \(A_j(t)\) is selected according to a normal distribution of standard deviation 3. The field changes every 100Â s.
To train the spin glass, we construct a drive by constructing a set \(\{A, B, \ldots \}\) of random fields. We randomly select a field from the set, then apply the field for 100Â s. This selectionandapplication process is performed 300 times (Fig.Â 2a).
The spin glass exchanges heat with a bath at a temperature \(T = 1 / \beta\). We set Boltzmannâ€™s constant to \(k_\text{B}= 1\). Energies are measured in Kelvins (K). To flip, a spin must overcome a heightB energy barrier. Spin j tends to flip at a rate \(\omega _j = e^{\beta [ H_j(t)  B]} / (1 \text { s}).\) This rate has the form of Arrheniusâ€™s law and obeys detailed balance. The average spin flips once per \(10^7\)Â s. We model the evolution with discrete 100s time intervals, using the Gillespie algorithm.
The spins absorb work when the field changes, as from \(\{ A_j(t) \}\) to \(\{ A'_j(t) \}\). The change in the spin glassâ€™s energy equals the work absorbed by the spin glass: \(W := \sum _{j = 1}^N\left[ A'_j(t)  A_j(t) \right] s_j.\) Absorbed power is defined as \(W / ( \text {100~s} )\). The spin glass dissipates heat by losing energy as spins flip.
The spin glass is initialized in a uniformly random configuration \({\mathcal {C}}\). Then, the spins relax in the absence of any field for 100,000Â s. The spin glass navigates to near a local energy minimum. If a protocol is repeated in multiple trials, all the trials begin with the same configuration \({\mathcal {C}}\).
In a certain parameter regime, the spin glass learns its drive effectively, even according to the absorbed power^{10}. Consider training the spin glass on a drive \(\{ A, B, C \}\). The spin glass absorbs much work initially. If the spin glass learns the drive, the absorbed power declines. If a dissimilar field D is then applied, the absorbed power spikes. If the familiar fields are reapplied, the absorbed power spikes again, but less. The spin glass learns effectively in the â€śGoldilocks regimeâ€ť of \(\beta = 3\)Â K\(^{1}\) and \(B = 4.5\)Â K^{10}: The temperature is high enough, and the barriers are low enough, that the spin glass can explore phase space. But T is low enough, and the barriers are high enough, that the spin glass is not hopelessly peripatetic.
Spins can fail to learn nontrivially, yet adopt configurations that reflect a drive. For example, the spins can be entrained to the field. The spins would bear the fieldâ€™s stamp as silly putty bears a thumbprint. A thumbprint vanishes as soon as the silly putty is smoothed. Hence the silly putty undergoes no longlived structural change that resists erasure; the silly putty does not learn robustly. Alternatively, most of the spins can remain frozen, while only a few flip. One might infer the drive from the few flippable spins, though most of the glass would contain no information about the drive. We confirm that our spin glass does not exhibit these behaviors, in Supplementary NoteÂ III: the spin glassâ€™s learning is nontrivial.
How to quantify a manybody systemâ€™s learning of a drive, using representation learning
We detect and quantify four facets of learning: classification ability, memory capacity, discrimination, and novelty detection. One classifies a stimulus by answering the question â€śWhich of the possible stimuli is this one?â€ť A systemâ€™s memory capacity is the number of fields that the system can remember. (We use the term â€śmemory capacityâ€ť in the physical sense of Ref.^{9}. A more specific, technical definition of â€śmemory capacityâ€ť is used in reservoir computing^{19}.) One performs novelty detection by answering the question â€śHave I encountered this stimulus before?â€ť One discriminates between stimuli A and B by answering â€śHow much of the present stimulus consists of A, and how much consists of B?â€ť
Below, we illustrate the application of our toolkit by quantifying classification ability. The Methods show how to apply our toolkit to the other three facets of learning. Further facets may be quantified similarly. Our machinelearning approach detects and measures learning more reliably and precisely than absorbed power does. Code used and data generated are accessible at Ref.^{20}.
A system classifies a stimulus when identifying the stimulus as one of many possibilities. First, we detail the protocol run on the spin glass. Second, we show how to measure the spin glassâ€™s classification ability using representation learning. Third, we measure the spin glassâ€™s classification ability using absorbed power. The neural network, we find, reflects more of the spin glassâ€™s classification ability than absorbed power does.
The spin glass underwent the following protocol. We generated random fields A, B, C, D, and E. From 4 of the fields, we formed the drive \({\mathcal {D}}_1 := \{A, B, C, D\}\). On the drive, we trained the spin glass in each of 1000 trials. In each of 1000 other trials, we trained a refreshed spin glass on a drive \({\mathcal {D}}_2 := \{A, B, C, E\}\). We repeated this process for each of the 5 possible 4field drives. Ninety percent of the trials were randomly selected for training the neural network. The rest were used for testing.
We measured the spin glassâ€™s ability to classify drives, using the neural network, as follows: We fixed a time t, then identified the configurations occupied by the spin glass at t in the spinglasstraining trials. On these configurations, we trained the neural network. The neural network populated the latent space with dots (similarly to in Fig.Â 3). The dots generated by drive \({\mathcal {D}}_j\) approximated a probability density \(P_j\), for all \(j = 1, 2, 3, 4, 5\).
We then gave the neural network a timet configuration from a test trial. The neural network compressed the configuration into a latentspace point. We calculated the probability that drive \({\mathcal {D}}_j\) generated that point, using \(P_j\), for all j. The highestprobability drive most likely generated the point, by maximumlikelihood estimation^{21}. We performed this testing and estimation for each trial in the test data. The fraction of trials in which the estimation succeeded constitutes the score. The score is plotted against t in Fig.Â 4 (blue, upper curve).
We compare with the classification ability attributed to the spin glass by the absorbed power: We fixed a drive \({\mathcal {D}}_j\) and a time t. We identified the neuralnetworktraining trials in which \({\mathcal {D}}_j\) was applied at time t. From the power absorbed then, we formed a histogram. We performed this process for each drive \({\mathcal {D}}_j\). Then, we took a trial from the test set and identified the power absorbed at t. We inferred which drive most likely produced that power, applying maximumlikelihood estimation to the histograms. The guessâ€™s score appears as the orange, lower curve in Fig.Â 4.
A score maximizes at 1.00 if the drive is always guessed accurately. The score is lowerbounded by the randomguessing value \(1 / (\text {number of drives}) = 1/5\). In Fig.Â 4, each score grows over tens of field switches. The absorbedpower score begins at 0.20 and comes to fluctuate around 0.25. The neural networkâ€™s score comes to fluctuate slightly below 1.00. (The neural networkâ€™s score begins slightly above 0.20. One might expect the score to begin at 0.20: At \(t = 0\), the spin glass has not experienced the drive, so the neural network receives no information about the drive, so the neural network can guess the drive only randomly. The distance from 0.20, we expect, comes from stochasticity of three types: the spin glassâ€™s initial configuration, maximumlikelihood estimation, and stochastic gradient descent. Stochasticity of only the first two types affects the absorbedpower score.) Hence the neural network detects more of the spin glassâ€™s classification ability than the absorbed power does, in addition to suggesting a means of quantifying the classification ability rigorously. Having illustrated our machinelearning toolkit with classification, we detail applications to memory capacity, novelty detection, and discrimination in the Methods.
Discussion
We have detected and quantified a manybody systemâ€™s learning of drive patterns, using representation learning. Our toolkit affords greater sensitivity than absorbed power, a representative of the thermodynamic toolkit applied to detect manybody learning until now. Our technique quantifies classification ability, memory capacity, discrimination ability, and novelty detection. The toolkit is general, not relying on whether the system exhibits magnetization or strain or another thermodynamic response. The Methods establish the feasibility of applying our toolkit in a variety of experiments and simulations. This approach provides a framework for understanding memoryâ€”a basic, widely realized, and usable traitâ€”in a unified manner across classical statistical mechanics. This framework opens several opportunities for future research; we detail two below.
First, our toolkit is wellsuited to more open problems about manybody learners. An example problem concerns the soapbubble raft in Ref. ^{7}. Experimentalists trained a raft of soap bubbles with an amplitude\(\gamma _\text{t}\) strain. The soap bubblesâ€™ positions were tracked, and variances in positions were calculated. No such measures distinguished trained rafts from untrained rafts; only stressing the raft and reading out the strain could^{7,22}. Our bottleneck neural network is wellpoised to identify microscopic properties that distinguish trained from untrained rafts. Similarly, representation learning may facilitate the detection of active matter. Selforganization is detected now through simple, largescale, easily visible signals^{23}. Bottleneck NNs could identify patterns invisible in thermodynamic measures.
Second, in statistical mechanics, we parameterize macrostates with volume, energy, magnetization, and other thermodynamic variables. Macrostates in statistical mechanics parallel the latent space in our bottleneck neural network (Fig.Â 1). Which variables parameterize the neural networkâ€™s latent space? Latent space may suggest definitions of new thermodynamic variables, or hidden relationships amongst known thermodynamic variables.
We illustrate by training the spin glass with a drive \(\{ A, B, C \}\) in each of many trials. On the endoftrial configurations, we trained the neural network. Two latentspace directions have physical significances, as shown in Fig.Â 5: the absorbed power grows along the diagonal from the bottom righthand corner to the upper lefthand corner (Fig.Â 5a). The magnetization grows radially (Fig.Â 5b). The directions are nonorthogonal, suggesting a nonlinear relationship between the thermodynamic variables. Convention biases physicists toward measuring volume, magnetization, heat, work, etc. The neural network may identify new macroscopic variables bettersuited to farfromequilibrium statistical mechanics, or nonlinear relationships amongst thermodynamic variables.
We can translate, as follows, between conventional thermodynamic variables and the latentspace directions \(z_1\) and \(z_2\): List the conventional thermodynamic variables expected to be relevant: \(v_1, v_2, \ldots , v_n\). For example, \(v_1\) may denote the work absorption, and \(v_2\) may denote the magnetization. The neural network populates the latent space with dots during training. Each dot corresponds to \(v_j\)â€™s calculable from the corresponding manybody configuration. A feedforward neural network can decompose each \(z_k\) as a function of the \(v_j\)â€™s. We will have decomposed the latentspace variables in terms of thermodynamic variables, translating between the two. A bottleneck neural network could uncover new theoretical physics, as discussed in, e.g., Refs.^{24,25,26}.
Methods
In the Results, we applied our machinelearning toolkit to quantify classification ability. Here, we apply the toolkit to quantify three more facets of learning: memory capacity, discrimination, and novelty detection. We also demonstrate the feasibility of applying our toolkit to experiments.
Memory capacity: How many fields can the system remember?
How many fields can a manybody system remember? A bottleneck neural network, we find, registers a greater memory capacity than absorbed power registers. Hence the neural network reflects statistical mechanical learning, at high field numbers, that the absorbed power does not.
We illustrate by constructing 50 random fields. We selected 40 to form a drive \({\mathcal {D}}_1\), selected 40 to form a drive \({\mathcal {D}}_2\), and repeated until forming five drives. We trained the spin glass on drive \({\mathcal {D}}_j\) in each of 1000 trials, for each of \(j = 1, 2, \ldots 5\). Ninety percent of the trials were designated as neuralnetworktraining trials; and 10%, as neuralnetworktesting trials.
The choice of 50 fields is explained in SupplementaryÂ NoteÂ IV: 50 fields exceed the spinglass capacity registered by the absorbed power. We will show that 50 fields do not exceed the capacity registered by the neural network: The neural network identifies spinglass learning missed by the absorbed power.
We used representation learning to quantify the spin glassâ€™s capacity as follows. For a fixed time t, we collected the configurations occupied by the spin glass at t in the neuralnetworktraining trials. On these configurations, the neural network performed unsupervised learning. The neural network populated its latent space with dots that formed five clusters. The cluster sourced by drive \({\mathcal {D}}_j\) approximated a probability density \(P_j\). We fed the neural network the configuration occupied at t during a test trial. The neural network formed a new dot in latent space. We estimated the probability that drive \({\mathcal {D}}_j\) formed the drive, using \(P_j\), for each j. The greatest probability stemmed from the drive \({\mathcal {D}}_j\) that most likely, according to the neural network, produced the point. That is, we applied maximumlikelihood estimation. The fraction of test trials in which the neural network guessed correctly constitutes the neural networkâ€™s score. The score is plotted against t in Fig.Â 6, as the blue, upper curve.
The neural networkâ€™s score is compared with the absorbed powerâ€™s score, calculated as follows. For a fixed time t, we identified the power absorbed at t in each neuralnetworktesting trial. We histogrammed the power absorbed when \({\mathcal {D}}_j\) was applied at t, for each \(j = 1, 2, \ldots , 5\). We then identified the power absorbed at t in a test trial. Comparing with the histograms, we inferred which drive was most likely being applied. We repeated this inference with each other test trial. In which fraction of the trials did the absorbed power identify the drive correctly? This number forms the absorbed powerâ€™s score. The score is plotted as the lower, orange curve in Fig.Â 6.
The higher the score, the greater the memory capacity attributed to the spin glass. The absorbed power identifies the drive in approximately \(20\%\) of the trials, as would random guessing. The score remains approximately constant, because the number of fields exceeds the spinglass capacity measured by the absorbed power. In contrast, the neural networkâ€™s score grows over \(\approx 150\) changes of the field, then plateaus at \(\approx 0.450\). The neural network points to the wrong drive most of the time but succeeds significantly more often than the absorbed power. Hence representation learning uncovers more of the spin glassâ€™s memory capacity than absorbed power measure does.
In summary, a manybody systemâ€™s memory capacity can be quantified as the greatest number of fields in any drive on which maximumlikelihood estimation, based on a neural networkâ€™s latent space, scores better than random guessing.
Discrimination: How new is this field?
Suppose that a manybody system learns fields A and B, then encounters a field that interpolates between them. Can the system recognize that the new field contains familiar constituents? Can the system discern how much A contributes and how much B contributes? The answers characterize the systemâ€™s discrimination ability, which we quantify with a maximumlikelihoodestimation score. Estimates formed from the neural networkâ€™s latent space reflect more of the systemâ€™s discriminatory ability than do estimates formed from absorbed power.
We illustrate with the spin glass, forming a drive \(\{A, B, C\}\). Each trial began with 300 subsequent time intervals. In each interval, a field was selected randomly from the drive and applied. The spin glass was then tested with a linear combination \(D_w = w A + (1  w) B\). The weight w varied from 0 to 1, in steps of 1/6, across trials.
We measured the spin glassâ€™s discrimination using the neural network as follows. We identified the final configuration assumed by the spin glass in each trial. These configurations were split into neuralnetworktraining data and neuralnetworktesting data. The training trials ended with configurations on which the neural network was trained. Then, the neural network received a configuration with which a neuralnetworktesting trial ended. The neural network mapped the configuration to a latentspace point. We inferred which field most likely generated that point, using maximumlikelihood estimation on the latent space. We tested the neural network with all the test trials. The fraction of maximumlikelihood estimates that were correct formed the neural networkâ€™s score.
Similarly, we measured the spin glassâ€™s discrimination using the absorbed power. We fixed a value of w, then identified the neuralnetworktraining trials that ended with the application of \(D_w\). We identified the power \({\mathcal {P}}\) absorbed by the spin glass after the \(D_w\) application. We histogrammed \({\mathcal {P}}\), inferring the probability that, if shown \(D_w\) for a given w, the spin glass will absorb an amount \({\mathcal {P}}\) of power. We formed a histogram for each value of w. Then, we calculated the power absorbed during a neuralnetworktesting trial. We inferred which field most likely generated that point, applying maximumlikelihood estimation to the histograms. We repeated the maximumlikelihood estimation with each neuralnetworktesting trial. The absorbed powerâ€™s score equals the fraction of the trials in which the maximumlikelihood estimation was correct.
The neural networkâ€™s score equals about double the absorbed powerâ€™s score, for latent spaces of dimensionality 2â€“20. The neural network scores between 0.448 and 0.5009, whereas the absorbed power scores 0.2381. Hence the representationlearning model picks up on more of the spin glassâ€™s discriminatory ability than the absorbed power does.
In summary, a manybody systemâ€™s ability to discriminate amongst combinations of familiar fields can be quantified with the score of maximumlikelihood estimates formed from a neural networkâ€™s latent space.
Novelty detection: Has the system encountered this drive before?
At the start of the introduction, we described how absorbed power has been used to identify novelty detection. A system detects novelty when labeling a stimulus as familiar or unfamiliar. The stimulus produces a response that exceeds a threshold or lies below. If the stimulus exceeds the threshold, an observer should guess that the stimulus is novel. Otherwise, the observer should guess that the stimulus is familiar.
The observer can err in two ways: One commits a false positive by believing a familiar drive to be novel. One commits a false negative by believing a novel drive to be familiar. The errors trade off: Raising the threshold lowers the probability \(p( {\text {pos.}}{\text {neg.}})\), suppressing false positives at the cost of false negatives. Lowering the threshold lowers the probability \(p({\text {neg.}}{\text {pos.}})\), suppressing false negatives at the cost of false positives.
The receiveroperatingcharacteristic (ROC) curve depicts the tradeoffâ€™s steepness (see Ref.^{27} and Fig.Â 7). Each point on the curve corresponds to one threshold value. The falsepositive rate \(p( {\text {pos.}}{\text {neg.}} )\) runs alongthe xaxis; and the truepositive rate, \(p( {\text {pos.}}{\text {pos.}})\), along the yaxis. The greater the area under the ROC curve, the more sensitively the response reflects accurate novelty detection.
We measure a manybody systemâ€™s noveltydetection ability using an ROC curve. Let us illustrate with the spin glass. We constructed two random drives, \(\{A, B, C \}\) and \(\{D, E, F \}\). We trained the spin glass on \(\{A, B, C \}\). In each of 3,000 trials, we then tested the spin glass with A, B, or C. In each of 3000 other trials, we tested with D, E, or F. We defined one response in terms of a bottleneck neural network, as detailed below; measured the absorbed power; and, from each response, drew an ROC curve (Fig.Â 7). The curves show that representation learning and absorbed power detect the spin glass's novelty detection about equally well. Each method excels slightly in one regime or another.
We defined the representationlearning response as follows. We trained the neural network on the configurations assumed by the spin glass during its training. The neural network populated latent space with three clumps of dots. We modeled the clumps with a hard mixture \(p_{ABC} (z_1, z_2)\) of three Gaussians. (A mixture is hard if it models each point as belonging to only one Gaussian.) We then fed the neural network the configuration that resulted from testing the spin glass. The neural network mapped the configuration to a latentspace point \((z_1^\text{test}, z_2^\text{test})\). We calculated the probability \(p_{ABC} (z_1^\text{test}, z_2^\text{test}) \, dz_1 dz_2\) that the ABC distribution produced the new point. This probability was compared to a fixed threshold. If the probability exceeded the threshold, the test configuration was guessed to have been produced by a novel drive. We repeated this protocol with the other test trials, using the fixed threshold. The fraction of guesses that were true positives, and the fraction of guesses that were false positives, specified one point on the blue, solid curve in Fig.Â 7. Varying the threshold led to the other points.
We defined a thermodynamic ROC curve in terms of absorbed power. Consider the trials in which the spin glass is tested with field A. We histogrammed the power absorbed by the spin glass after the A test. We formed another histogram from the Btest trials; and a third histogram, from the Ctest trials. To these histograms was compared the power \({\mathcal {P}}\) that the spin glass absorbed during a test with an arbitrary field. We inferred the likelihood that \({\mathcal {P}}\) resulted from a familiar field. The results form the orange, dashed curve in Fig.Â 7.
The two ROC curves enclose regions of approximately the same area: the neural network curve encloses an area0.9633 region; and the thermodynamic curve, an area0.9601 region. On average across all thresholds, therefore, the responses register novelty detection approximately equally. Yet the responses excel in different regimes: The neural network achieves greater truepositive rates at low falsepositive rates, and the absorbed power achieves greater truepositive rates at high falsepositive rates. This tworegime behavior persisted across batches of trials, though the enclosed areas fluctuated slightly. Hence anyone paranoid about avoiding false positives should measure a manybody systemâ€™s novelty detection with a neural network. Those more relaxed might prefer the absorbed power.
Why should the neural network excel at low falsepositive rates? Because of the neural networkâ€™s skill at generalizing, we expect. Upon training on cat pictures, a neural network generalizes from the instances. Shown a new cat, the neural network recognizes its catness. Perturbing the input a little perturbs the neural networkâ€™s response little. Hence changing the magnetic field a little, which changes the spinglass configuration little, should change latent space little, obscuring the manybody systemâ€™s novelty detection. This obscuring disappears when the magnetic field changes substantially.
In summary, a manybody systemâ€™s noveltydetection ability is quantified with an ROC curve formed from a neural networkâ€™s latent space or a thermodynamic response, depending on the falsepositive threshold.
Feasibility
Applying our toolkit might appear impractical, since microstates must be inputted into the neural network. Measuring a manybody systemâ€™s microstate may daunt experimentalists. Yet the use of microstates hinders our proposal little, for three reasons.
First, microstates can be calculated in numerical simulations, which inform experiments. Second, many key properties of manybody microstates have been measured experimentally. Highspeed imaging has been used to monitor soap bubblesâ€™ positions^{7} and colloidal suspensions^{28}. Similarly wielded tools, such as high magnification, have advanced activematter^{29} and geneexpression^{30} studies.
One might worry that the full microstate cannot be measured accurately or precisely. Soap bubblesâ€™ positions can be measured with finite precision, and other microscopic properties might be inaccessible. But, third, some bottleneck neural networks denoise their inputs^{12,31}: The neural networks learn the distribution from which samples are generated ideally, not systematic errors. Denoising by variational autoencoders is less established but is progressing^{32}.
Data availability
The machinelearning and spinglasssimulation code is available at Ref.^{20}. Will be available at^{20} once COVID19 restrictions loosen enough that we can access the computers that store the files.
References
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Acknowledgements
The authors thank Alexander Alemi, Isaac Chuang, Emine Kucukbenli, Nick Litombe, Seth Lloyd, Julia Steinberg, Tailin Wu, and Susanne Yelin for useful discussions. WZ is supported by ARO Grant W911NF1810101; the Gordon and Betty Moore Foundation Grant, under No. GBMF4343; and the Henry W. Kendall (1955) Fellowship Fund. JMG is funded by the AFOSR, under Grant FA99501710136. SM was supported partially by the Moore Foundation, via the Physics of Living Systems Fellowship. This material is based upon work supported by, or in part by, the Air Force Office of Scientific Research, under award number FA95501910411. JLE has been funded by the Air Force Office of Scientific Research Grant FA95501710136 and by the James S. McDonnell Foundation Scholar Grant 220020476. NYH is grateful for an NSF grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and the Smithsonian Astrophysical Observatory. NYH also thanks CQIQC at the University of Toronto, the Fields Institute, and Caltechâ€™s Institute for Quantum Information and Matter (NSF Grant PHY1733907) for their hospitality during the development of this paper.
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J.M.G. simulated the spin glass. W.Z. built the machinelearning code and processed the spinglass data. S.M. built the noveltydetection code. N.Y.H. managed the project and wrote the manuscript. All authors (W.Z., J.M.G., S.M., J.L.E., and N.Y.H.) contributed to the project design and data analysis. W.Z. and J.M.G. contributed equally.
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Zhong, W., Gold, J.M., Marzen, S. et al. Machine learning outperforms thermodynamics in measuring how well a manybody system learns a drive. Sci Rep 11, 9333 (2021). https://doi.org/10.1038/s41598021883117
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DOI: https://doi.org/10.1038/s41598021883117
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