Model learning device, model learning method, and program

Abstract

According to the present invention, a processor of a model training device comprises an input data processing unit, a model setting processing unit, a parameter estimation unit, and a parameter processing unit. The input data processing unit causes a storage to store input data. The model setting processing unit sets, in a deep generative model, a structure based on model setting. The parameter estimation unit inputs input data into the model that has been trained by repeatedly feeding training data to the deep generative model having the set structure and estimates a parameter related to an oscillator. The parameter processing unit causes the storage to store the estimated parameter.

Description

[Technical Field]

[0001] FIELD OF THE INVENTION Embodiments of the present invention relate to techniques for training neural networks. [Background technology]

[0002] The existence of phenomena in which individual rhythms synchronize with other surrounding rhythms has been widely confirmed, such as the light emission of fireflies, the vocalizations of frogs, human walking, and applause in a concert hall. Phase oscillator models (sometimes abbreviated as phase models), such as the Kuramoto model described below, are widely used as models to represent such collective synchronization phenomena.

[0003] The phase oscillator model involves an entity called an oscillator that moves rhythmically. The phase oscillator model describes the progress of the oscillator's rhythm using phase. In the above example, one firefly, frog, or human corresponds to one oscillator. Phase is a value that indicates which state an oscillator is in in its periodic light emission, vocalization, or walking rhythm. The phase oscillator model is defined by expressing, in the form of a differential equation, how forces act on each other among multiple oscillators and how this affects the time evolution of phase.

[0004] Many mathematical properties of the phase oscillator model have been investigated, including its asymptotic behavior. For example, in the Kuramoto model, the order parameter, which determines the degree of synchronization, converges depending on the value of the coupling strength parameter, which determines the degree of force transmitted between oscillators.

[0005] Phase oscillator models are increasingly recognized as playing a central role in the analysis of synchronization phenomena. However, their use in modeling actually observed synchronization phenomena is not easy. This is because phase oscillator models generally include unknown parameters in differential equations, and the phase itself is generally unobservable, so both the parameters and the phase must be estimated. For these reasons, the use of phase oscillator models is currently limited to cases where the phase can be determined with some degree of accuracy from actual measurements. For example, Non-Patent Document 1 discloses an approach based on the Poincaré map or the Hilbert transform. [Prior art documents] [Non-patent literature]

[0006] [Non-Patent Document 1] Arkady Pikovsky, Michael Rosenblum, Jurgen Kurz, and Isao Tokuda (translators). Fundamentals and Applications of Synchronization Theory: From Mathematical Sciences, Chemistry, and Life Sciences to Engineering, pp. 149-167, Maruzen, 2009. Summary of the Invention [Problem to be solved by the invention]

[0007] The Poincaré mapping approach is a method for determining phase for data that has an explicit marker. For example, an electrocardiogram has an event (R peak) that is observed once per cycle. This can be used as a marker to set the reference phase time (the time at which the phase becomes zero), and the phase at each time can be determined by linearly interpolating the time between markers. However, this approach cannot be applied to data that does not have a marker. Even if a marker exists, if the marker detection fails, the phase cannot be determined correctly. For example, it is well known that electrocardiogram waveforms acquired by wearable devices when the user is not in a resting state are contaminated by noise such as body movement, and R peak detection often fails. It is highly likely that the phase cannot be accurately determined for such data.

[0008] The Hilbert transform is a method for determining phase using the concept of an analytic signal. Formally, it can determine the (instantaneous) phase and instantaneous amplitude from any signal. However, the Hilbert transform is sensitive to low-frequency trends, and preprocessing such as filtering is required. Furthermore, although phase should increase linearly with time, it has been reported that the Hilbert transform can cause the (instantaneous) phase to oscillate. It has been pointed out that the phase obtained by this method should only be treated as an estimate.

[0009] As such, each approach has its own challenges, and whichever approach is adopted, it is necessary to consider individual processing methods for each case, such as marker detection methods and preprocessing methods, for each piece of data to be analyzed. The present invention has been made in light of the above circumstances, and aims to provide a technique that makes it possible to simultaneously estimate phase and parameters and to investigate the phenomenon to be analyzed from the results. [Means for solving the problem]

[0010] A model learning device according to one aspect of the present invention is a model learning device that trains a deep generative model having the phase of an oscillator as a hidden variable. The model learning device includes a storage that stores a program, a memory into which the program is loaded from the storage, and a processor that processes information according to instructions written in the program loaded into the memory. The storage includes an input data storage unit that stores a time series of observation values ​​as input data, a model setting storage unit that stores model settings for the deep generative model, and a parameter storage unit that stores the output of the deep generative model. The processor includes an input data processing unit, a model setting processing unit, a parameter estimation unit, and a parameter processing unit. The input data processing unit stores input data in the storage. The model setting processing unit sets a structure based on the model settings in the deep generative model. The parameter estimation unit inputs input data to the deep generative model, which has been trained by repeatedly providing training data to the structured deep generative model, and estimates parameters related to the oscillator. The parameter processing unit stores the estimated parameters in the storage. [Effects of the Invention]

[0011] According to one aspect of the present invention, it is possible to simultaneously estimate the phase and the parameters, and to investigate the phenomenon to be analyzed from the results. [Brief explanation of the drawings]

[0012] [Figure 1] FIG. 1 is a diagram showing an example of a system to which a model learning device according to an embodiment of the present invention is applied. [Figure 2] FIG. 2 is a block diagram showing an example of the model learning device 1 shown in FIG. [Figure 3] FIG. 3 is a functional block diagram showing an example of the model learning device 1 shown in FIG. [Figure 4] FIG. 4 is a flowchart showing an example of a processing procedure performed by the model learning device 1. [Figure 5] FIG. 5 is a diagram showing an example of the flow of data in the model learning device 1. DETAILED DESCRIPTION OF THE INVENTION

[0013] Hereinafter, an embodiment of the present invention will be described with reference to the drawings.

[0014] <About the Kuramoto Model> The Kuramoto model is disclosed in, for example, references [6] and

[13] .

[0015] <Summary> The inventors have constructed a new model in which the phase of each oscillator is used as a hidden variable. This model is called the Latent Phase Oscillator Model (LPOM). LPOMs are classified as deep generative models. LPOMs include at least three neural networks: an encoder network, a phase oscillator network, and a decoder network. In an embodiment, the encoder network receives a time series of observations as input and outputs parameters of a probability distribution that the phase value at the start time of the time series follows. The phase oscillator network solves the differential equation of the phase oscillator model. The decoder network receives a phase value at each time and outputs parameters of a probability distribution that the observation value follows.

[0016] Like widely used neural networks such as CNNs, LPOM can estimate latent variables, including phase, and the parameters of each network based on data using backpropagation. This allows for simultaneous estimation of both phase and parameters without preprocessing each analysis data. The estimated phase can be used to visualize the phase value at each time. It can also be used to calculate a quantity representing the degree of synchronization (such as the order parameter in the Kuramoto model), which is calculated using the phase. This makes it possible to investigate the phenomenon under analysis. Furthermore, the estimated phase can also be used to predict the phase and observed values ​​at future times.

[0017] LPOM is not a model specialized for analyzing a specific type of data, but can be used for general purposes to analyze various types of data. For example, since any (phase-related) differential equation model can be used for a phase oscillator network, it is possible to create a phase oscillator network based on a well-known model such as the Kuramoto model. It is also possible to consider an extended version of the Kuramoto model in which a term corresponding to an external force (not handled in the standard Kuramoto model) is added to the differential equation. Furthermore, it is also possible to use models other than the Kuramoto model.

[0018] Furthermore, LPOM can estimate phases and parameters even when data is incomplete, such as when observations are missing. For example, consider a case where only observations at a specific time are available as data. Even in such cases, by designing LPOM to calculate the loss function used to estimate phases and parameters by calculating the error between the observations at that observation time and the model output, estimation is possible even with incomplete data.

[0019] Synchronous phenomena have been observed in a variety of domains, including the aforementioned light emission of fireflies, the vocalizations of frogs, human walking, and applause in concert halls, as well as the movement of metronomes, candle flames, human circadian rhythms, heartbeats, and cells. Even for phenomena whose existence has not been demonstrated at the time of filing, for example, the existence of synchronous phenomena in applause in concert halls can be easily inferred to exist in the waving of glow sticks (penlights). In other words, LPOM is not limited to the above examples and can be used universally regardless of the type of synchronous phenomenon to be analyzed.

[0020] <Preparation> (Phase oscillator model) First, we will explain the phase oscillator model (references [6], [9]) that forms the basis of the method described in the embodiments. A phase oscillator model (phase model) is a model in which the state of an oscillator is described only by the phase φ, and the time evolution (dynamics) of the phase φ is expressed by a differential equation. A quasi-linear oscillator, one of the simplest phase oscillators, is derived from an oscillator (1) in which the position of the oscillator changes over time according to a sinusoidal wave. (2) in equation (1) is the phase, and the phase dynamics can be expressed by a differential equation (4) using a function (3) that takes the phase as input.

[0021] In addition, the position y of the oscillator when the phase is φ is determined as (6) using the function (5) that takes the phase as input, so the state of the oscillator can be expressed only by the phase. sinSince is a sine function, phases that differ by a constant multiple of 2π, for example, φ(t), φ(t)+2π, and φ(t)+4π, can all be considered to be the same. In the following explanation, phases that differ by a constant multiple of 2π will be considered to be the same. In general, phases that differ by a constant multiple of the constant T can also be considered to be the same.

[0022] In the phase oscillator model, there are many models that express the fact that multiple oscillators are weakly coupled and exert forces on each other. The phase of the i-th oscillator at time t is φ i (t), or simply φ i As with the quasi-linear phase oscillator above, phases that differ by a constant multiple of 2π can be considered to be the same. The total number of oscillators is assumed to be N, and all phases are collectively represented by the symbol 《7》.

[0023]

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[0024] In the Kuramoto model, a typical example of a phase oscillator model, the phase dynamics is expressed by the (nonlinear) differential equation of Equation (1).

[0025]

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[0026] However, w in equation (1) i is the natural frequency, and K is a parameter that represents the coupling strength between oscillators. It is easy to imagine that this can express synchronization phenomena from the following example.

[0027] Let's assume that the total number of oscillators is N=2, and that at a certain time t, the phase φ1 of oscillator 1 is slightly ahead of the phase of oscillator 2, i.e., the phase difference ε=φ1-φ2 ​​satisfies 0≦ε≦π. In this case, since the phase is 《8》, a force acts on oscillator 1 to slow down its effective (instantaneous) frequency 《9》. Conversely, since the phase is 《10》, a force acts on oscillator 2 to increase its effective frequency 《9》. Therefore, this model expresses the forces acting on each oscillator to align their frequencies. If the coupling strength K is large enough, after a sufficient amount of time has passed, the two oscillators will continue to operate at equal frequencies, operating in a synchronized (frequency) state.

[0028] As a parameter expressing the degree of synchronization, the quantity 《11》 expressed by equation (2), called the order parameter, is defined and used.

[0029]

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[0030]

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[0031] <12>, and when all oscillators are in phase, R(t) = 1. Since R(t) = 0 when the oscillators are uniformly spread over [0, 2π], it can be interpreted that the closer the value is to 1, the more synchronized the oscillators are. Therefore, it is common to simultaneously grasp the value of each phase and the degree of synchronization by displaying a diagram that visualizes the phase (using <13>) as a group of points on a unit circle centered at the origin and the center of gravity of that group of points (the absolute value of the center of gravity coincides with the order parameter).

[0032] This order parameter has been mathematically analyzed, and it has been shown that when the total number of oscillators N is infinite and the coupling strength K is below a certain threshold, R(t) approaches 0 after sufficient time has passed, but when it is above the threshold, it approaches a value greater than 0 (a value that increases as K increases).

[0033] So far, we have used the Kuramoto model for explanation. Based on the above discussion, it is easy to extend it to a model with higher expressive power. The following four are examples of extended models:

[0034] [(A) Adding a function representing external forces] For example, adding a function u(t) representing a time-dependent external force gives us equation (3).

[0035]

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[0036] [(B) Using general periodic functions] If we consider using a general periodic function Γ instead of a sine function, we obtain equation (4).

[0037]

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[0038] [(C) When parameters / functions that were common among oscillators are made oscillator-dependent parameters / functions] For example, let us consider the coupling strength K as an oscillator-dependent parameter K ij If we express it as:

[0039]

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[0040] (D) When a time-independent parameter is made a time-dependent parameter (i.e., a function) In this case, we obtain equation (6).

[0041]

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[0042] The extension methods are not limited to those described above, and it is also possible to combine these methods or add external information (such as information about the position or attributes of the oscillators). In the following explanation, the extended Kuramoto model will also be referred to as the Kuramoto model.

[0043] (Neural ODE) The phase oscillator models explained so far have been described using differential equations. The method according to the embodiment also uses a phase oscillator model. For parameter estimation, for example, an approach called Neural Ordinary Differential Equation (Neural ODE) proposed in Reference [2] can be used.

[0044] Neural ODEs have recently been used in models described by (ordinary) differential equations. Here, we explain parameter estimation for a model in which the dynamics of a certain variable z(t) (not necessarily a phase) is expressed as a function h with a parameter θ, as in 14.

[0045]

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[0046] Generally, model parameter estimation involves minimizing the sum (called the loss function) of the error between the observed values ​​actually obtained at several times and the values ​​estimated by the model. Commonly used loss functions include cross entropy, absolute error, squared error, Huber loss, KL divergence, and Bregman divergence. However, the discussion here does not depend on the loss function used.

[0047] Let us denote the error calculated from the observed value at time t1 and the model estimated value z(t1) as L(z(t1)). Since z(t1) is given as the solution to a differential equation, this error can be expressed by the following equation (100).

[0048]

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[0049] However, the symbol ODESolve in equation (100) represents the operation of finding the value z(t1) at time t1 by solving a differential equation when the initial value at the initial time t0 is z(t0) and the function h defines the dynamics with the parameter θ.

[0050] Minimizing the loss function requires the gradient 15 with respect to the parameter θ of the model. Neural ODE efficiently calculates the gradient by introducing the adjoint variable 16.

[0051]

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[0052] Specifically, the adjoint variables themselves follow the differential equation (equation (101)) below, and are calculated by solving this differential equation backward from time t1 to time t0. Furthermore, the gradient of the desired parameter is calculated using relational equation (102).

[0053]

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[0054] In this embodiment, the above procedure naturally incorporates the operation of solving differential equations into the neural network, making it possible to estimate parameters. Various evolutionary methods have also been proposed for Neural ODE (references [1], [7]). These evolutionary methods can also be used for parameter estimation of the proposed LPOM.

[0055] <About the Latent Phase Oscillator Model (LPOM)>

[0056]

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[0057] Next, we will explain the proposed latent phase oscillator model (LPOM). This model includes three neural networks: an encoder network, a phase oscillator network, and an observation network. The phase of each oscillator and the parameters in the model are estimated by minimizing a loss function calculated using the model output and observation data, similar to parameter estimation in a normal neural network. In the following discussion, the observation data is estimated at certain times t0, t1, ..., t M (These are t0 <t1,…,<t M The explanation will be given assuming that the set of observed values ​​of (all) oscillators in the time series {17} and {18} is given in the form of sequence data. M represents the total number of sets. The same can be done when multiple sequence data are given.

[0058] (Data generation process) LPOM is defined as a deep generative model that treats phase as a latent variable. The observed data 《19》 is expressed as generated as follows:

[0059] (Step 1) The phase 20 at time t0 is generated according to a certain probability distribution 21 (equation (103)).

[0060]

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[0061] This probability distribution can be any one. In deep generative models, it is often assumed to be the (N-dimensional) standard normal distribution (Equation (104)), as in [5].

[0062]

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[0063] In equation (104), 《22》 is an N-dimensional vector with all elements zero, and 《23》 is an N×N identity matrix. μ and Σ are the mean and covariance matrices. Although it is possible to use the normal distribution mentioned above in this model, considering that the phase is a periodic variable, it is more natural to use the von Mises distribution, which takes values ​​on the [0, 2π] scale, as in reference [4].

[0064]

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[0065] The von Mises distribution is uniform when 24 and approximately normal when 25 is large. As shown in [3] and

[10] , it is also possible to use a method based on the multidimensional von Mises-Fisher distribution, modified by a transformation to polar coordinates.

[0066] (Step 2) The phase 《20》 obtained in step 1 is used as the initial value, and a (periodic) function f (the parameter of this function is θ) that determines the phase dynamics 《26》 is calculated. fThe details of the function f will be described later.) M The phase value 27 at is determined (by solving the differential equation).

[0067]

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[0068] In equation (106), the symbol δ represents a delta function.

[0069] Equation (106) is the time from t0 to t M Solve the differential equations up to time t1,t2,…,t M If we redefine the symbol ODESolve to return values ​​at multiple times, these solutions can also be written as follows:

[0070]

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[0071] (Step 3) In step 3, at each time t m The parameter θ conditioned on the phase 《29》 obtained in step 2 for each g The observed value 《30》 shown in equation (107) is generated according to the conditional probability distribution with the phase 《29》 as input and the parameter of the conditional probability distribution is called a decoder network. Details will be explained later.

[0072]

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[0073] For example, if the oscillator is a metronome, y corresponds to the position of the pendulum, and if the oscillator is a firefly, y corresponds to the intensity of the light emitted by the firefly.

[0074] The phase at each time is represented by the symbol 《31》 and the observation value is represented by the symbol 《32》. M The product of these probability distributions in is written as equation (108).

[0075]

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[0076] By steps 1, 2, and 3, the joint probability distribution of the phase and the observation value when the parameter is 《33》 is given by equation (8).

[0077]

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[0078] Since the phase Φ is a latent variable, we can marginalize and eliminate it to obtain the probability distribution of equation (9).

[0079]

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[0080] Note that the symbol 《34》 in the equation is 《36》 obtained by substituting the solution of the differential equation (solved with the initial value 《35》) for phases other than the initial time. The logarithm of this equation, 《37》, is a value called the log-likelihood, and is used in designing the objective function.

[0081] (Objective function) The objective function for estimating the latent variables and parameters of the above model can use the quantity 38 called the variational lower bound in equation (109), similar to the Variational Auto Encoder (VAE) (Reference [5]).

[0082]

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[0083] The conditional probability distribution can be the same as the initial phase probability distribution《21》. For example, if a normal distribution is used, a neural network with parameter λ that outputs the mean and variance, which are parameters of a Gaussian distribution, from the input Y can be used (this neural network is called an encoder network; details will be described later). If the mean and variance, which are the outputs of this neural network, are written as《40》 and《41》, respectively, the conditional probability distribution《39》 can be written as in equation (110).

[0084]

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[0085] Similarly, if the von Mises distribution (or von Mises-Fisher distribution) is used for the probability distribution of the initial phase, this conditional probability distribution can be expressed using the von Mises distribution, as in (References [3], [4],

[10] ). Note that the reason why "38" is called the variational lower bound is because it is a quantity obtained as a lower bound on the log-likelihood using Jensen's inequality, as in equation (111).

[0086]

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[0087] The variational lower bound 《38》 can be expanded as in equation (10).

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[0089] KL represents the KL-divergence between distributions, which can be calculated analytically if both distributions are set to be normal or von Mises distributions (References [4], [5]). When using normal distributions, the first term on the right-hand side of equation (10) can be approximated as equation (11) by using the reparameterization trick (Reference [5]).

[0090]

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[0091] where {42} is a value sampled from the (multidimensional) standard normal distribution {43}, and {44} is the symbol for the product of elements. Usually, L is set to 1. Therefore, we define the approximate value {45} of the variational lower bound as follows, and estimate the parameters λ and θ by maximizing this.

[0092]

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[0093] To maximize the variational lower bound, we need to minimize the loss function defined by inverting the sign of the variational lower bound. Any optimization method used in neural networks can be used for this calculation, such as (stochastic) steepest descent, (stochastic) Newton's method, or Adam.

[0094] When the steepest descent method is used, the parameters are repeatedly updated using equation (13).

[0095]

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[0096] When multiple (D) time series 《46》 are given as observed data, the objective function of equation (14) can be minimized in the same way as above.

[0097]

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[0098] When using the von Mises distribution, it is possible to minimize the objective function in the same way as above by using a method called the implicit reparameterization trick (Reference [4]) or a method using rejection sampling (Reference [8]).

[0099] Based on the above discussion, once the neural network structures of the encoder network, phase oscillator network, and decoder network (hyperparameter settings such as the total number of hidden layers, the method of inter-layer connections, and the activation function to be used) are determined, it is possible to estimate the parameters of these networks from observed data. Next, we will explain the details of each network.

[0100] (encoder network) The encoder network is a neural network that outputs the parameters of the probability distribution {39} that the phase {35} follows at time t0 from the observed data Y. If a Gaussian distribution is used as the probability distribution, {47} and {48} are the outputs of the encoder network.

[0101]

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[0102] Any neural network structure that can take time series data as input can be used, including multi-layer perceptrons (MLPs), convolutional neural networks (CNNs), recurrent neural networks (RNNs), long-short-term memories (LSTMs), and combinations of these networks.

[0103] Since the output is related to the phase at the initial time t0, when using RNN, the final observation time tm We often use RNNs that take a reverse time series《49》starting from the observed value of . The same is true when dealing with other neural networks.

[0104]

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[0105] (Phase oscillator network) A phase oscillator network is a neural network that expresses a (periodic) function f that determines the time evolution of the phase used in equation (7), and outputs the solution of the differential equation that follows that function f. The Kuramoto model or its extended model can be used for function f, or a phase oscillator model other than the Kuramoto model, or any model defined so that f is a periodic function, can also be used.

[0106] The reason why f needs to be a periodic function is because the input is a phase. Therefore, in the following explanation, we do not limit the form of the model, but let f be the function that determines the dynamics of the phase vector 《7》 consisting of N oscillators, and let θ be the parameter included in the model. f That is, we consider a phase oscillator model expressed by the dynamics shown in equation (15).

[0107]

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[0108] In addition, the parameter θ fIn addition to the parameters shown in the Kuramoto model (AD), we also assume that includes parameters such as neural networks (NNs). This is because, when considering time-dependent parameters such as A and D, it is natural to express the functions u(t) and K(t) as NNs that take time t as input. Any neural network can be used as the NN, including multilayer perceptrons (MLPs), radial basis function networks (RBF networks), convolutional neural networks (CNNs), recurrent neural networks (RNNs), and long-short-term memory (LSTMs).

[0109] (Decoder Network) The decoder network calculates the m In this case, the neural network takes the phase 《29》 as input and outputs the parameters of the probability distribution 《50》 that the observed value 《30》 follows. When using a normal distribution as the probability distribution as in the encoder network, a normal distribution such as equation (113) can be used, in which the output of the decoder network g is 《51》.

[0110]

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[0111]

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[0112] Depending on the nature of the observed values, it is possible to use any probability distribution other than the normal distribution. For example, if the observed values ​​are only discrete values, it is possible to use the negative binomial distribution or the Poisson distribution. Other possibilities include probability distributions that belong to the exponential distribution family or probability distributions that can express multimodality, such as mixture distributions.

[0113] However, unlike the encoder network, the input of the decoder network is phase (just like the phase oscillator network), so the decoder network must be designed to be a periodic function with the same period as the phase oscillator network. For example, if the phase oscillator network is represented by a sine function (with a period of 2π) based on the Kuramoto model, the decoder network must also be a periodic function with a period of 2π.

[0114] Furthermore, if it is necessary to give the same meaning to the phase values ​​between each oscillator (for example, so that a phase of zero corresponds to the smallest or largest value of light or sound intensity) in order to interpret the estimation results, it is necessary to design a structure such as an activation function so that the decoder network has such characteristics.

[0115] Let us consider a concrete example. Let us analyze the synchronization of the blinking of lights in a group of fireflies. For simplicity, let us say that each firefly periodically blinks its light, and at time t, when the i-th firefly (oscillator) is emitting light, the time y i (t)=1 is observed, and at time t′ when no light is emitted, y i Suppose (t′)=0 is observed.

[0116] "Designing a decoder network so that the phase value of each oscillator has the same meaning" means, for example, that the phase value is zero φ i = 0 (or more precisely, the value of zero plus a constant multiple of 2π), the state y i If we use a decoder network that does not consider this condition, the i-th firefly will output φ i =0 is the point when the light goes out (y i =0), and for the jth firefly, φ j =π / 2 at the time when the light goes out (y j= 0), and the meaning of the phase value may change for each oscillator. However, even in such cases, it is possible to perform an analysis that does not use the phase value of each oscillator itself, for example, to analyze whether the phase difference or effective (instantaneous) frequency is aligned, so it is not a requirement to design a decoder network with the meaning of phase.

[0117] A concrete example of a decoder network that takes into account the above-mentioned phase significance is the model 《53》 shown in equation (16), which expresses the observed value as a linear sum of L trigonometric functions (sine functions) with parameter 《52》 and a constant term (dependent on the oscillator).

[0118]

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[0119] However, the symbol [ ] i represents the i-th component of the vector. By using trigonometric functions, 《53》 becomes a periodic function. In addition, the parameter 《54》 related to the trigonometric functions is used in common for each oscillator, so the meaning of the phase of each oscillator can be unified.

[0120] Another example is a model (58) such as equation (17) with parameters (57) calculated using a neural network (56) common to oscillators with parameters (55) (independent of the oscillator index i) and oscillator-dependent constant and amplitude terms, after inputting the input phase into sine and cosine functions.

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[0122] Any neural network can be used for the neural network 56, such as a multilayer perceptron (MLP), a convolutional neural network (CNN), or a combination of these networks. In the above example, the observed value y i We have explained the network that represents the mean value of the predicted distribution by the model. The same network structure can also be used for the variance value of 60. Furthermore, the observed value y i A similar network structure can be used even when is expressed as a vector value rather than a scalar.

[0123]

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[0124] Based on the above discussion, an embodiment of the present invention will now be described.

[0125] <Embodiment> (composition) Fig. 1 is a diagram showing an example of a system to which a model learning device according to an embodiment of the present invention is applied. The system shown in Fig. 1 is a system in which an AI model 3 is trained by a model learning device 1, and the results are provided to an external device 2 for operation.

[0126] 2 is a block diagram showing an example of the model learning device 1 shown in FIG. The model learning device 1 is a computer including a processor 100, a memory 20, a storage 50, an input / output unit 60, and a bus 45 interconnecting these components. The input / output unit 60 establishes communication links between the model learning device 1, an external device 2, and an AI model 3, and transmits and receives various types of data. The processor 100 is a computing device such as a CPU (Central Processing Unit) or an MPU (Micro Processing Unit), and realizes the processing functions of the embodiment in accordance with a program loaded from the storage 50 into the memory 20.

[0127] Fig. 3 is a functional block diagram showing an example of the model learning device 1 shown in Fig. 2. In Fig. 3, memory 20 is a semiconductor memory such as a ROM (Read Only Memory) or a RAM (Random Access Memory).

[0128] The storage 50 is a non-volatile memory such as a hard disk drive (HDD) or a solid state drive (SSD), and stores a program 54 for implementing the processing according to the embodiment in addition to basic software such as an operating system (OS). That is, the program 54 can be installed in the model learning device 1. The storage 50 also includes an input data storage unit 51, which is a storage area for input data, a model setting storage unit 52, which is an area for storing model settings of the deep generative model related to the embodiment, and a parameter storage unit 53, which is a storage area for estimated parameters.

[0129] The processor 100 includes an input data processing unit 11, a model setting processing unit 12, a parameter estimation unit 13, and a parameter processing unit 14 as processing functions according to an embodiment of the present invention. The input data processing unit 11, the model setting processing unit 12, the parameter estimation unit 13, and the parameter processing unit 14 are realized by the processor 100 executing a program loaded into the memory 20. In other words, the program 54 includes instructions for causing the processor 100 to function as the input data processing unit 11, the model setting processing unit 12, the parameter estimation unit 13, and the parameter processing unit 14.

[0130] The input data processing unit 11 stores the input data in the storage 50. The model setting processing unit 12 sets, in the deep generative model, a structure based on the model settings stored in the model setting storage unit 52. The LPOM as a deep generative model has a structure including an encoder network that receives a time series of observation values ​​as input and outputs parameters of a probability distribution followed by the phase value at the start time of the time series, a phase oscillator network that solves the differential equation of the phase oscillator model, and a decoder network that receives an input of the phase value at each time and outputs parameters of the probability distribution followed by the observation value.

[0131] The parameter estimation unit 13 inputs the input data stored in the storage 50 into a model that has been trained by repeatedly providing training data to the LPOM with the above structure set. As a result, the trained LPOM outputs an estimation result of the oscillator parameters.

[0132] The parameter processing unit 14 stores the estimated parameters in the storage 50.

[0133] (action) Next, the operation of the above configuration will be described. Fig. 4 is a flowchart showing an example of a processing procedure by the model learning device 1. In Fig. 4, the processor 100 inputs the observation data <<19>> and stores it in the input data storage unit 51 of the storage 50 (step S1). Next, the processor 100 sets the structures of the encoder network, the phase oscillator network, and the decoder network in the neural network of the LPOM (step S2). Next, the processor 100 provides input data to the trained LPOM and estimates the parameters 61 of the model (step S3).

[0134]

number

[0135] Next, the processor 100 outputs the parameter θ of the model (step S4).

[0136] 5 is a diagram showing an example of the flow of data in the model learning device 1. For example, data is processed according to the following procedures (1) to (4). (1) The input data processing unit 11 stores the observation data in the input data storage unit 51. (2) The model setting processing unit 12 stores in the model setting storage unit 52 information such as the neural network structures of the encoder network, the phase oscillator network, and the decoder network, and the loss function used for estimation. (3) The parameter estimation unit 13 receives the observed data 19 from the input data storage unit 51 and the information from the model setting storage unit 52, minimizes the objective function of equation (12) using the (stochastic) steepest descent method shown below, and obtains the parameters 62 of each neural network. Then, the parameters λ and θ are stored in the parameter storage unit 53.

[0137]

number

[0138] (a) Initialize the parameters λ and θ. Set the maximum number of iterations as a variable used as the termination condition. (b) The parameters λ and θ are updated according to equation (13). (c) Update the number of calculation iterations. (d) If the number of calculation iterations exceeds the predetermined number of iterations, the process ends. If not, the process returns to (b). (4) The parameter processing unit 14 outputs the parameters λ and θ in the parameter storage unit 53 to an external device.

[0139] The operation of each component of the model learning device 1 according to the embodiment can be constructed as a program, which can be installed and executed on a computer used as the model learning device, or distributed via a network. The present invention is not limited to the above-described embodiments, and various modifications and applications are possible.

[0140] (effect) According to the embodiment, it is possible to estimate latent variables including phase and parameters of each network based on data, visualize the value of phase at each time, and calculate quantities that represent the degree of synchronization calculated using phase, such as the order parameter in the Kuramoto model, a representative phase oscillator model, thereby enabling the phenomenon under analysis to be investigated.Furthermore, it is also possible to predict the values ​​of phase and observations at future times.

[0141] Existing techniques include methods for determining phase for data with explicit markers or methods for determining phase using the concept of an analytic signal. However, these methods cannot be applied to data without markers. Furthermore, the properties of the phase obtained may differ from the properties of the original phase, making it difficult to accurately determine the phase.

[0142] In contrast, in this embodiment, learning involving error backpropagation is repeated for a model including three neural networks: an encoder network that receives a time series of observation values ​​as input and outputs parameters of a probability distribution to which the phase value at the start time of the time series conforms; a phase oscillator network that solves the differential equation of a phase oscillator model; and a decoder network that receives a phase value at each time as input and outputs parameters of a probability distribution to which the observation value conforms. Using a trained model trained using this method, latent variables including phase and the parameters of each network can be estimated based on data. In other words, it becomes possible to simultaneously estimate both phase and parameters, and ultimately to investigate the phenomenon under analysis using a quantity (such as an order parameter) that represents the degree of synchronization calculated using the estimated phase. From these facts, according to the embodiment, it is possible to provide a technique that makes it possible to simultaneously estimate phases and parameters and to investigate the phenomenon to be analyzed from the results.

[0143] The present invention is not limited to the above-described embodiment. For example, in the embodiment, the steepest descent method is used to estimate the parameters λ and θ that minimize the objective function, but it is also possible to use a (stochastic) Newton method, a stochastic gradient method, Adam, etc. Furthermore, it is also possible to use a probability distribution other than the normal distribution or the von Mises distribution for the encoder and decoder networks.

[0144] In addition, although one time series Y is used in the embodiment, multiple time series 63 may be used. The objective function can be expressed by equation (14). The same method as described above can be used to minimize the objective function.

[0145]

number

[0146] Furthermore, the functional units of the model learning device 1 may be distributed across multiple devices, and these devices may perform processing in cooperation with each other. Each functional unit may also be realized using a circuit. The circuit may be a dedicated circuit that realizes a specific function, or may be a general-purpose circuit such as a processor.

[0147] Furthermore, the flow of each process described above is not limited to the procedures described, and the order of some steps may be changed, or some steps may be performed simultaneously in parallel. Furthermore, the series of processes described above do not need to be performed consecutively, and each step may be performed at any timing.

[0148] Processor 100 may also be implemented in a variety of other forms, including integrated circuits such as application specific integrated circuits (ASICs) and field-programmable gate arrays (FPGAs).

[0149] The program for realizing the above-described processing may be provided by being stored on a computer-readable recording medium (or storage medium). The program is stored on the recording medium as an installable file or an executable file. Examples of recording media include magnetic disks, optical disks (CD-ROM, CD-R, DVD-ROM, DVD-R, etc.), magneto-optical disks (MO, etc.), and semiconductor memories. The program for realizing the above-described processing may also be stored on a computer (server) connected to a network such as the Internet and downloaded to a computer (client) via the network.

[0150] In addition, the specific types of algorithms and the configuration of the neural network can be modified in various ways without departing from the spirit of the present invention.

[0151] The model learning device according to the embodiment can be configured by constructing the operation of each component as a program, which can be installed and executed on a computer used as the model learning device, or distributed via a network. The present invention is not limited to the above-described embodiments, and various modifications and applications are possible.

[0152] In short, this invention is not limited to the above-described embodiments, and various modifications can be made in the implementation stage without departing from the spirit of the invention. Furthermore, the embodiments may be implemented in appropriate combinations, in which case the combined effects can be obtained. Furthermore, the above-described embodiments include various inventions, and various inventions can be extracted by combining selected elements from the disclosed elements. For example, if the problem can be solved and the desired effect can be obtained even if some elements are deleted from all elements shown in the embodiments, the configuration from which these elements are deleted can be extracted as an invention.

[0153] <References> [1] Ricky TQ Chen, Brandon Amos, and Maximilian Nickel. Learning neural event functions for ordinary differential equations. arXiv preprint arXiv:2011.03902, 2020. [2] RickyTQ Chen, Yulia Rubanova, Jesse Bettencourt, and DavidK Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems, Vol.31, , 2018. [3] TimR Davidson, Luca Falorsi, Nicola DeCao, Thomas Kipf, and Jakub M Tomczak. Hyperspherical variational auto-encoders. In Uncertainty in Artificial Intelligence, pp. 856-865, 2018. [4] Mikhail Figurnov, Shakir Mohamed, and Andriy Mnih. Implicit reparameterization gradients. Advances in Neural Information Processing Systems, Vol.31, , 2018. [5] Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013. [6] Yoshiki Kuramoto. Chemical oscillations, waves, and turbulence. Springer-Verlag, 1984. [7] Takashi Matsubara, Yuto Miyatake, and Takaharu Yaguchi. Symplectic adjoint method for exact gradient of neural ode with minimal memory. Advances in Neural Information Processing Systems, Vol.34, , 2021. [8] Christian Naesseth, Francisco Ruiz, Scott Linderman, and David Blei. Reparameterization gradients through acceptance-rejection sampling algorithms. In Artificial Intelligence and Statistics, pp. 489-498. PMLR, 2017. [9] ArthurT Winfree. The geometry of biological time, Vol.2. Springer, 1980.

[10] Jiacheng Xu and Greg Durrett. Spherical latent spaces for stable variational autoencoders. In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pp. 4503-4513, 2018.

[12] Hiroshi Gun. The spread of research on synchronization phenomena (Keywords of modern physics). Journal of the Physical Society of Japan, Vol. 69, No. 9, pp. 602-603, 2014.

[13] Yuki Kuramoto. The Synchronized World of Nonlinear Science. Shueisha, 2014. [Explanation of symbols]

[0154] 1...Model learning device 2...External device 3. AI model 11...Input data processing unit 12...Model setting processing section 13...Parameter estimation unit 14...Parameter processing section 20...Memory 45...Bus 50…Storage 51...input data storage unit 52...Model setting memory section 53...Parameter storage unit 54...Program 60...Input / output section 100...Processor.

Claims

1. A model learning device that learns a deep generative model having a phase of an oscillator as a hidden variable, a storage device for storing a program; a memory into which the program is loaded from the storage; a processor for processing information according to instructions written in a program loaded into said memory; The storage an input data storage unit that stores a time series of observation values ​​as input data; a model setting storage unit that stores model settings of the deep generative model; a parameter storage unit that stores an output of the deep generative model; The processor: an input data processing unit that stores the input data in the storage; a model setting processing unit that sets a structure based on the model setting to the deep generative model, the model setting processing unit including an encoder network that receives the time series of the observation values ​​as input and outputs parameters of a probability distribution to which the phase value at the start time of the time series follows, a phase oscillator network that solves a differential equation of a phase oscillator model, and a decoder network that receives the phase value at each time as input and outputs parameters of a probability distribution to which the observation value follows; a parameter estimation unit that inputs the input data into a model that has been trained by repeatedly providing training data to the deep generative model having the set structure, and estimates parameters related to the oscillator; a parameter processing unit that stores the estimated parameters in the storage.

2. A model learning device as described in Claim 1, wherein the probability distribution output by the encoder network is a Gaussian distribution or a von Mises distribution.

3. A model learning method for training a deep generative model having oscillator phases as hidden variables by a computer having a storage for storing a program, a memory into which the program is loaded from the storage, and a processor for processing information in accordance with instructions written in the program loaded into the memory, comprising: the processor storing a time series of observations as input data in the storage; the processor storing model settings of the deep generative model in the storage; the processor storing an output of the deep generative model in the storage; The processor sets a structure based on the model setting in the deep generative model, the structure including an encoder network that receives the time series of the observation values ​​as input and outputs parameters of a probability distribution to which the phase value at the start time of the time series follows, a phase oscillator network that solves a differential equation of a phase oscillator model, and a decoder network that receives the phase value at each time as input and outputs parameters of a probability distribution to which the observation value follows; A model learning method comprising: the processor inputting the input data into a model that has been trained by repeatedly providing training data to the deep generative model with the set structure, and estimating parameters related to the oscillator.

4. A program including instructions for causing the processor to execute processing by each part of the device described in either claim 1 or claim 2.

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