Trajectory data prediction device, trajectory data prediction method, and program

Abstract

A trajectory data prediction device according to one embodiment of the present disclosure has: a parameter learning unit that is configured to learn learnable parameters including parameters of Gaussian processes to which a vector field conforms, the vector field expressing a physical system in which conservation of energy or dissipation of energy is established, so as to maximize a probability distribution to which trajectory data conforms, the trajectory data representing the trajectory of a physical quantity observed in the physical system, and the learning being carried out on the basis of the similarity of the Gaussian processes; and a predicted trajectory data calculation unit that is configured to predict trajectory data satisfying the prediction condition by using the vector field, the prediction being made on the basis of the learnable parameters that have been learned and associated prediction conditions.

Description

[Technical Field]

[0001] The present disclosure relates to a trajectory data prediction device, a trajectory data prediction method, and a program. [Background technology]

[0002] In recent years, with the development and spread of sensing technology, various pieces of information from the real world are being accumulated as data. By utilizing such data to analyze natural phenomena in various fields such as physics, chemistry, and ecology, attention is being drawn to the possibility of discovering new laws of nature and making predictions about the future. Hereinafter, a pair of the observation time and the physical quantity when some physical quantity is observed in the real world is considered to be one sample, and the sequence of such samples is referred to as "trajectory data."

[0003] A method has been proposed for predicting the time evolution of physical quantities by using machine learning techniques with trajectory data as input (e.g., Non-Patent Document 1). Machine learning techniques incorporating prior knowledge in physics (such as the law of conservation of energy and the law of dissipation) have also been proposed (e.g., Non-Patent Documents 2 to 5), which have enabled more accurate prediction of trajectory data. [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] RTQ Chen, Y. Rubanova, J. Bettencourt, and DK Duvenaud. Neural ordinary differential equations.In Advances in Neural Information Processing Systems, volume 31, 2018. [Non-patent document 2] S. Greydanus, M. Dzamba, and J. Yosinski. Hamiltonian neural networks. In Advances in Neural Information Processing Systems, volume 32, 2019. [Non-patent document 3] K. Rath, CG Albert, B. Bischl, and U. von Toussaint. Symplectic Gaussian process regression of maps in Hamiltonian systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 31(5):053121, 2021. [Non-patent document 4] YD Zhong, B. Dey, and A. Chakraborty. Dissipative SymODEN: Encoding Hamiltonian dynamics with dissipation and control into deep learning. In ICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations, 2020. [Non-Patent Document 5] YD Zhong, B. Dey, and A. Chakraborty. Symplectic ODE-Net: Learning Hamiltonian dynamics with control. In International Conference on Learning Representations, 2020. Summary of the Invention [Problem to be solved by the invention]

[0005] The conventional methods described in Non-Patent Documents 2, 4, and 5 incorporate the theory of Hamiltonian mechanics into a neural network, making it possible to predict trajectory data that obeys the laws of conservation and dissipation of energy. However, these conventional methods implicitly assume that a sufficient amount of training data with high time resolution is available. Therefore, if such data is not available, overfitting to the data occurs, resulting in a problem of reduced prediction performance.

[0006] On the other hand, the conventional method described in Non-Patent Document 3 incorporates the theory of Hamiltonian mechanics into the Gaussian process model, a machine learning method. Gaussian process models are generally expected to prevent overfitting even when a sufficient amount of data is not available. However, because this conventional method performs training based on finite difference approximations calculated from trajectory data, there is a problem in that when the time resolution of the trajectory data is low, the error of the finite difference approximation increases, resulting in a decrease in prediction performance. Furthermore, this conventional method only targets the law of conservation of energy and cannot be applied to phenomena involving energy dissipation.

[0007] The present disclosure has been made in consideration of the above points, and provides a technology for predicting trajectory data using a Gaussian process model that can handle both the law of energy conservation and the law of dissipation. [Means for solving the problem]

[0008] A trajectory data prediction device according to one aspect of the present disclosure includes: a parameter learning unit configured to learn learnable parameters including parameters of a Gaussian process, based on an approximation of a Gaussian process followed by a vector field representing a physical system in which the law of conservation of energy or the law of energy dissipation holds, so as to maximize a probability distribution followed by trajectory data representing a trajectory of a physical quantity observed in the physical system; and a predicted trajectory data calculation unit configured to predict trajectory data that satisfies a prediction condition using the vector field, based on the learned learnable parameters and a given prediction condition. [Effects of the Invention]

[0009] A technique for predicting trajectory data is provided using a Gaussian process model that can handle both the laws of energy conservation and dissipation. [Brief explanation of the drawings]

[0010] [Figure 1] FIG. 1 is a diagram illustrating an example of a hardware configuration of a trajectory data prediction device according to an embodiment of the present invention. [Figure 2] FIG. 2 is a diagram illustrating an example of a functional configuration of the trajectory data prediction device according to the present embodiment. [Figure 3] 10 is a flowchart showing an example of the flow of processing executed by the trajectory data prediction device according to the present embodiment. [Figure 4] FIG. 10 is a diagram illustrating an example of a predicted trajectory data visualization screen. DETAILED DESCRIPTION OF THE INVENTION

[0011] An embodiment of the present invention will be described below. A method for predicting trajectory data using a Gaussian process model that can handle both the law of conservation of energy and the law of dissipation is proposed. Furthermore, a parameter learning method utilizing an ordinary differential equation (ODE) solver is proposed to enable model learning directly from trajectory data provided as training data without using finite difference approximation. Furthermore, an approximation method using randomized Fourier features is also proposed to improve the efficiency of this learning.

[0012] Below, the above proposed method will be explained, and then a trajectory data prediction device 10 that predicts trajectory data using this proposed method will be explained.

[0013] <Proposed method> This proposed method can be applied to all natural phenomena.

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[0015] In this proposed method, a generative model for trajectory data is constructed using a Gaussian process incorporating the theory of Hamiltonian mechanics, and its unknown parameters are estimated based on the variational Bayes method, after which trajectory data is predicted for any prediction conditions (initial conditions representing the initial state and prediction time representing the time to develop from the initial state).

[0016] Assume that the following dataset has been obtained as a training dataset.

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[0019] ≪Parameter estimation≫ First, we construct a generative model for trajectory data using a Gaussian process incorporating the theory of Hamiltonian mechanics. Next, we assume that the training data was obtained according to this generative model, and perform parameter learning (estimation) using the variational Bayes method.

[0020] Derivation of Gaussian process models incorporating the theory of Hamiltonian mechanics Based on the idea of ​​Hamiltonian mechanics, we define the Hamiltonian (total energy of the system) as

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[0025] Following the procedure for deriving the Hamiltonian dynamical system, we define the differential operator

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[0033] Since differentiation is a linear operator, it is known that the derivative of a Gaussian process is also a Gaussian process. Therefore, equation (4) becomes

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[0037] Equations (6) and (7) are Gaussian processes that can handle both the law of conservation of energy and the law of dissipation of energy, and this Gaussian process model is used as a generation model for trajectory data.

[0038] Approximation method using randomized Fourier features It is possible to calculate the predictive distribution from equation (6) and develop a learning algorithm using an ODE solver, but this would require a huge amount of calculation. Therefore, we consider approximating equation (6) using randomized Fourier features (e.g., Reference 1).

[0039] First, we approximate equation (2) with M pairs of basis functions. The m-th pair of basis functions is

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[0048] Equation (10) is equivalent to an approximation of equation (6) using randomized Fourier features. By using this equation (10), the value of the vector field at any point x can be efficiently evaluated, enabling parameter learning using the ODE solver described below.

[0049] Parameter learning using ODE solvers We will explain how to learn (estimate) various parameters of the generative model for trajectory data. First, according to the concept of a probabilistic generative model, we estimate the observed value y ij We model the generation process of data (hereinafter referred to as observed data) consisting of observed data Y = {y ij} is assumed to be generated according to the following probability distribution:

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[0052] In this case, we want to estimate the parameters so as to maximize Equation (12), but it is difficult to analytically calculate Equation (12) because it involves solving a differential equation. Therefore, we will derive a learning algorithm using the Variational Bayes method below.

[0053] In the variational Bayes method, we use Jensen's inequality to determine the lower bound for the logarithm of Eq. (12) as

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[0058] Using Monte Carlo integration, we calculate the expectation value in Eq. (14) as

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[0063] Under the above conditions, various parameters are estimated so as to maximize the right-hand side of equation (14). Any continuous optimization method can be used for this maximization.

[0064] <Trajectory data prediction> Initial conditions (x q ,x p ) and the predicted time are given as prediction conditions. At this time, the trajectory data is predicted using various learned parameters. Hereinafter, the trajectory data to be predicted is also referred to as "predicted trajectory data," and a method for calculating the predicted trajectory data will be described.

[0065] Using the trained parameters, the predicted mean of the vector field f(x) is

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[0067] <Example of Hardware Configuration of Trajectory Data Prediction Device 10> An example of the hardware configuration of a trajectory data prediction device 10 according to this embodiment is shown in Fig. 1. As shown in Fig. 1, the trajectory data prediction device 10 according to this embodiment includes an input device 101, a display device 102, an external I / F 103, a communication I / F 104, a RAM (Random Access Memory) 105, a ROM (Read Only Memory) 106, an auxiliary storage device 107, and a processor 108. Each of these pieces of hardware is connected to each other via a bus 109 so as to be able to communicate with each other.

[0068] The input device 101 is, for example, a keyboard, a mouse, a touch panel, a physical button, etc. The display device 102 is, for example, a display, a display panel, etc. Note that the trajectory data prediction device 10 does not necessarily have to include at least one of the input device 101 and the display device 102, for example.

[0069] The external I / F 103 is an interface with an external device such as a recording medium 103a. The trajectory data prediction device 10 can read from and write to the recording medium 103a via the external I / F 103. Examples of the recording medium 103a include a flexible disk, a CD (Compact Disc), a DVD (Digital Versatile Disk), an SD memory card (Secure Digital memory card), and a USB (Universal Serial Bus) memory card.

[0070] The communication I / F 104 is an interface for connecting the trajectory data prediction device 10 to a communication network. The RAM 105 is a volatile semiconductor memory (storage device) that temporarily stores programs and data. The ROM 106 is a non-volatile semiconductor memory (storage device) that can store programs and data even when the power is turned off. The auxiliary storage device 107 is a storage device (storage device) such as an HDD (Hard Disk Drive), an SSD (Solid State Drive), or a flash memory. The processor 108 is an arithmetic device such as a CPU (Central Processing Unit) or a GPU (Graphics Processing Unit).

[0071] The trajectory data prediction device 10 according to this embodiment has the hardware configuration shown in Fig. 1 and can realize various processes described below. Note that the hardware configuration shown in Fig. 1 is an example, and the hardware configuration of the trajectory data prediction device 10 is not limited to this. For example, the trajectory data prediction device 10 may have multiple auxiliary storage devices 107 or multiple processors 108, may not have some of the hardware shown in the figure, or may have various hardware other than the hardware shown in the figure.

[0072] <Example of functional configuration of trajectory data prediction device 10> FIG. 2 shows an example of the functional configuration of the trajectory data prediction device 10 according to this embodiment. As shown in FIG. 2, the trajectory data prediction device 10 according to this embodiment includes a parameter estimation unit 201, a prediction condition receiving unit 202, a predicted trajectory data calculation unit 203, and an output unit 204. These units are realized, for example, by processing in which one or more programs installed in the trajectory data prediction device 10 are executed by the processor 108 or the like. The trajectory data prediction device 10 according to this embodiment also includes a trajectory data storage unit 301, a kernel parameter storage unit 302, a friction coefficient parameter storage unit 303, a noise variance parameter storage unit 304, and a variation parameter storage unit 305. These units are realized, for example, by the auxiliary storage device 107.

[0073] However, for example, all or part of the trajectory data storage unit 301, the kernel parameter storage unit 302, the friction coefficient parameter storage unit 303, the noise variance parameter storage unit 304, and the variation parameter storage unit 305 may be realized by a storage device, a database server, or the like connected to the trajectory data prediction device 10 via a communication network.

[0074] The parameter estimation unit 201 uses the Gaussian process of equation (6) to assume a probability distribution (equation (12)) that the observed data follows, and then estimates (learns) various parameters so as to maximize the probability distribution using equation (10), which approximates this Gaussian process using randomized Fourier features, the variational Bayes method, and an ODE solver. That is, the parameter estimation unit 201 estimates (learns) the kernel parameter σ0 so as to maximize the right-hand side of equation (14). 2 and Λ, the friction coefficient parameter R, and the noise variance parameter σ 2 and the variational parameters A, b, and C are estimated.

[0075] The prediction condition receiving unit 202 receives the prediction conditions (initial conditions (x q ,x p ) and prediction time). Here, the prediction conditions may be specified, for example, by an input device 101 included in the trajectory data prediction device 10, or may be specified by a keyboard, mouse, touch panel, or the like included in a terminal connected to the trajectory data prediction device 10 via a communication network. Alternatively, for example, the prediction conditions may be specified by another program installed in the trajectory data prediction device 10, or may be specified by a program installed in a terminal connected to the trajectory data prediction device 10 via a communication network.

[0076] The predicted trajectory data calculation unit 203 calculates predicted trajectory data using the prediction conditions accepted by the prediction condition acceptance unit 202 and various learned parameters. That is, the predicted trajectory data calculation unit 203 calculates the prediction average of the vector field f(x) using equation (20), and then calculates predicted trajectory data using an arbitrary ODE solver using this prediction average and the initial conditions and prediction time included in the prediction conditions. In this way, predicted trajectory data representing the trajectory when the initial conditions are time-evolved by the prediction time is calculated.

[0077] The output unit 204 outputs the predicted trajectory data calculated by the predicted trajectory data calculation unit 203 to a predetermined output destination. Here, the predicted trajectory data may be output, for example, to the display device 102 included in the trajectory data prediction device 10, to the auxiliary storage device 107 included in the trajectory data prediction device 10, or to a display included in a terminal connected to the trajectory data prediction device 10 via a communication network. Alternatively, the predicted trajectory data may be output to a printer and printed on a printing medium such as paper, or output as sound from a speaker or the like, or may be output to another program installed in the trajectory data prediction device 10, or to a program installed in a terminal connected to the trajectory data prediction device 10 via a communication network, or to a database server connected to the trajectory data prediction device 10 via a communication network.

[0078] The trajectory data storage unit 301 stores a learning dataset (the dataset shown in formula (1)) provided to the trajectory data prediction device 10. Note that the trajectory data prediction device 10 may have, for example, an operation unit for performing various operations on the trajectory data storage unit 301. Here, the various operations may include, for example, an operation to modify or delete all or part of the dataset stored in the trajectory data storage unit 301, an operation to add data to the trajectory data storage unit 301, etc.

[0079] The kernel parameter storage unit 302 stores the learned kernel parameter σ0 2 and Λ.

[0080] The friction coefficient parameter storage unit 303 stores the friction coefficient parameter R learned by the parameter estimation unit 201.

[0081] The noise variance parameter storage unit 304 stores the learned noise variance parameter σ 2 Remember.

[0082] The variational parameter storage unit 305 stores the variational parameters A, b, and C that have been learned by the parameter estimation unit 201.

[0083] Note that the functional configuration of the trajectory data prediction device 10 shown in Fig. 2 is an example and is not limited to this. For example, although the trajectory data prediction device 10 has each unit in the example shown in Fig. 2, at least some of the units shown in Fig. 2 may be included in a device or equipment connected to the trajectory data prediction device 10 via a communication network.

[0084] <Flow of processing executed by trajectory data prediction device 10> The flow of processing executed by the trajectory data prediction device 10 according to this embodiment will be described below with reference to Fig. 3. Steps S101 to S102 in Fig. 3 are a parameter estimation phase for estimating (learning) various parameters, and steps S103 to S105 in Fig. 3 are a trajectory data prediction phase for calculating predicted trajectory data.

[0085] The parameter estimation unit 201 uses the learning data set (the data set shown in Equation (1)) stored in the trajectory data storage unit 301 to estimate various parameters (kernel parameters σ0 2 and Λ, the friction coefficient parameter R, and the noise variance parameter σ 2 , variational parameters A, b, and C are estimated (step S101).

[0086] Then, the parameter estimation unit 201 estimates (learns) the learned kernel parameter σ0 2 and Λ are stored in the kernel parameter storage unit 302, the learned friction coefficient parameter R is stored in the friction coefficient parameter storage unit 303, and the learned noise variance parameter σ 2 is stored in the noise variance parameter storage unit 304, and the learned variation parameters A, b, and C are stored in the variation parameter storage unit 305 (step S102).

[0087] The prediction condition receiving unit 202 receives the prediction conditions (initial conditions (x q ,x p ) and predicted time) are accepted (step S103).

[0088] Next, the forecasted trajectory data calculation unit 203 calculates forecasted trajectory data using the prediction conditions received in step S103 and the various learned parameters (step S104).

[0089] Then, the output unit 204 outputs the predicted trajectory data calculated in step S104 to a predetermined output destination (step S105). Here, as an example, a case will be described in which the predicted trajectory data is output to the display device 102, such as a display, and the trajectory represented by the predicted trajectory data is visualized on a predetermined screen (hereinafter, also referred to as a predicted trajectory data visualization screen). The predicted trajectory data visualization screen 1000 shown in FIG. 4 includes an initial condition specification field 1100, which is an input field in which the user can specify an initial condition, a predicted time specification field 1200, which is an input field in which the user can specify a predicted time, and a predicted trajectory visualization field 1300, in which a trajectory represented by the predicted trajectory data predicted from the initial condition and the predicted time specified in the initial condition specification field 1100 and the predicted time specification field 1200, respectively, is visualized. By specifying the desired initial conditions and predicted time in the initial condition specification field 1100 and the predicted time specification field 1200, respectively, on this predicted trajectory data visualization screen 1000, the user can see the trajectory (predicted trajectory) represented by the predicted trajectory data predicted from those initial conditions and predicted time from the predicted trajectory visualization field 1300.

[0090] <Summary> As described above, the trajectory data prediction device 10 according to this embodiment introduces a Gaussian process incorporating both the law of conservation of energy and the law of dissipation, and then uses this Gaussian process to assume a probability distribution to which the observation data will follow, and estimates various parameters so as to maximize this probability distribution. In addition, at this time, the trajectory data prediction device 10 according to this embodiment approximates the Gaussian process using randomized Fourier features, and estimates various parameters using the variational Bayes method and an ODE solver.

[0091] As a result, the trajectory data prediction device 10 according to this embodiment can predict trajectory data obtained by evolving a physical system over time when the initial conditions of the physical system are given. Therefore, by using the trajectory data prediction device 10 according to this embodiment, it becomes possible to analyze natural phenomena in physical systems in which energy is not only conserved but also dissipated, thereby making it possible to discover new laws of nature and predict the future, for example.

[0092] The present invention is not limited to the above-described specifically disclosed embodiments, and various modifications, changes, and combinations with known technologies are possible without departing from the scope of the claims.

[0093] [References] Reference 1: A. Rahimi and B. Recht. Random features for large-scale kernel machines. In Advances in Neural Information Processing Systems, volume 20, 2007. Reference 2: RTQ Chen, B. Amos, and M. Nickel. Learning neural event functions for ordinary differential equations. In International Conference on Learning Representations, 2021. [Explanation of symbols]

[0094] 10 Trajectory data prediction device 101 Input Device 102 Display device 103 External I / F 103a Recording media 104 Communication I / F 105 RAM 106 ROM 107 Auxiliary storage 108 processors 109 Bus 201 Parameter Estimation Unit 202 Prediction condition reception unit 203 Predicted trajectory data calculation unit 204 Output section 301 Trajectory data storage unit 302 Kernel parameter storage 303 Friction coefficient parameter memory 304 Noise variance parameter storage unit 305 Variational parameter storage

Claims

1. a parameter learning unit configured to learn learnable parameters including parameters of a Gaussian process, based on an approximation of a Gaussian process followed by a vector field representing a physical system in which the law of conservation of energy or the law of energy dissipation holds, so as to maximize a probability distribution followed by trajectory data representing a trajectory of a physical quantity observed in the physical system; and a predicted trajectory data calculation unit configured to predict trajectory data that satisfies a given prediction condition using the vector field based on the learned learnable parameters and a given prediction condition; A trajectory data prediction device having the following.

2. The parameter learning unit The trajectory data prediction device according to claim 1 , configured to approximate the Gaussian process using randomized Fourier features.

3. The parameter learning unit The trajectory data prediction device according to claim 2 , wherein the learnable parameters are learned using a variational Bayesian method and an ODE solver.

4. The trajectory data prediction device according to claim 1 , wherein the learnable parameters include at least a kernel parameter of the Gaussian process and a parameter for expressing the energy dissipation law.

5. The trajectory data prediction device according to claim 4, wherein the vector field is a mapping in which a Hamiltonian representing the total energy of the physical system is subjected to a differential operator having a coefficient adjusted by a parameter for representing the energy dissipation law.

6. a parameter learning procedure for learning learnable parameters including parameters of a Gaussian process, based on an approximation of a Gaussian process followed by a vector field representing a physical system in which the law of conservation of energy or the law of energy dissipation holds, so as to maximize a probability distribution followed by trajectory data representing a trajectory of a physical quantity observed in the physical system; a predicted trajectory data calculation step of predicting trajectory data that satisfies a given prediction condition using the vector field based on the learned learnable parameters and the given prediction condition; The computer executes the trajectory data prediction method.

7. a parameter learning procedure for learning learnable parameters including parameters of a Gaussian process, based on an approximation of a Gaussian process followed by a vector field representing a physical system in which the law of conservation of energy or the law of energy dissipation holds, so as to maximize a probability distribution followed by trajectory data representing a trajectory of a physical quantity observed in the physical system; a predicted trajectory data calculation step of predicting trajectory data that satisfies a given prediction condition using the vector field based on the learned learnable parameters and the given prediction condition; A program that causes a computer to execute the following.

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