Information processing device, information processing method, and program

Abstract

This information processing device comprises: a prediction unit that calculates an objective function by converting input information comprising observation information of a predetermined physical quantity and a control variable for the predetermined physical quantity into a latent variable of a low-dimensional latent space, identifying a state equation by sparse regression to which Regularization with Oracle is applied using the latent variable, and performing time-series prediction using the state equation; and a control unit that calculates optimal control by solving an optimization problem based on the objective function.

Description

Information processing device, information processing method, and program

[0001] This invention relates to a technology for controlling plasma in a nuclear fusion reactor.

[0002] SINDy (Non-Patent Document 1) is software that infers the governing equations (state equations) that a time-series dataset follows from observed information of a target dynamic system using sparse regression.

[0003] It has been proposed that model predictive control (MPC) of fusion plasma is possible by using the equation of state obtained from SINDy as a predictive model of the system and solving an optimization problem for the objective function within the sampling period based on that prediction (Non-Patent Literature 2).

[0004] Model-based predictive control consists of a prediction part that represents the controlled object using mathematical models, and a control part that calculates the optimal control by solving an optimization problem.

[0005] On the other hand, it is believed that transforming the input space into a lower-dimensional latent space can improve generalization performance, speed up learning, and suppress overfitting (Non-Patent Literature 3). A representative method for reducing the dimensionality of observed information is the autoencoder (AE), a type of neural network (Non-Patent Literature 4), which can transform a high-dimensional input space into a lower-dimensional latent space and extract features from the input space. Research combining AE with other machine learning methods is also being conducted in SINDy (Non-Patent Literature 5-7).

[0006] Representations obtained by transforming high-dimensional observational information into low-dimensional state variables are called reduced-order models (ROMs), and they are used in many fields such as machine learning, control engineering, and physics.

[0007] Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the national academy of sciences, 113(15):3932-3937, 2016.JD Lore, S De Pascuale, P Laiu, B Russo, J-S Park, JM Park, SL Brunton, JN Kutz, and AA Kaptanoglu. Time-dependent solps-iter simulations of the tokamak plasma boundary for model predictive control using sindy. Nuclear Fusion, 63(4):046015, 2023.Timothee Lesort, Natalia Diaz-Rodriguez, Jean-Franois Goudou, and David Filliat. State representation learning for control: An overview. Neural Networks, 108:379-392, 2018.Pierre Baldi and Kurt Hornik. Neural networks and principal component analysis: Learning from examples without local minima. Neural networks, 2(1):53-58, 1989.Kathleen Champion, Bethany Lusch, J Nathan Kutz, and Steven L Brunton. Data-driven discovery of coordinates and governing equations.Proceedings of the National Academy of Sciences, 116(45):22445-22451, 2019.Kai Fukami, Takaaki Murata, Kai Zhang, and Koji Fukagata. Sparse identification of nonlinear dynamics with low-dimensionalized flow representations. Journal of Fluid Mechanics, 926:A10, 2021.Joseph Bakarji, Kathleen Champion, J Nathan Kutz, and Steven L Brunton. Discovering governing equations from partial measurements with deep delay autoencoders. arXiv preprint arXiv:2201.05136, 2022.Jianqing Fan and Runze Li. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American statistical Association, 96(456):1348-1360, 2001.Peng Zheng, Travis Askham, Steven L Brunton, J Nathan Kutz, and Aleksandr Y Aravkin. A unified framework for sparse relaxed regularized regression: Sr3. IEEE Access, 7:1404-1423, 2018.

[0008] For model predictive control, it is necessary to correctly estimate the equation of state that a fusion plasma follows. For model predictive control, whether for low-dimensional or high-dimensional models, identifying the equation of state by sparse regression requires correctly estimating true non-zero elements and their coefficients. However, the sparse regularization methods implemented in SINDy, such as l0 regularization and l1 regularization, lack oracle properties (Non-Patent Literature 8).

[0009] This invention has been made in view of the above points, and aims to provide a technology that enables the correct estimation of the equation of state followed by a fusion plasma and the performance of model-based predictive control.

[0010] According to the disclosed technology, an information processing device is provided, comprising: a prediction unit that converts input information consisting of observational information of a predetermined physical quantity and control variables for the predetermined physical quantity into latent variables in a low-dimensional latent space, identifies a state equation by sparse regression with oracle-like regularization applied using the latent variables, and calculates an objective function by performing time series prediction using the state equation; and a control unit that calculates optimal control by solving an optimization problem based on the objective function.

[0011] The disclosed technology provides a method for correctly estimating the equation of state followed by a fusion plasma and enabling model predictive control.

[0012] This figure shows an example of the overall system configuration in an embodiment of the present invention. This figure shows the specific configuration of the system. This is a flowchart for explaining the operation of the information processing device 100. This figure shows a map when the plasma position change is converted into a two-dimensional latent space. This figure shows an example of predictive control in latent space performed by the information processing device 100. This figure shows an example of predictive control in latent space performed by the information processing device 100. This figure shows an example of predictive control in latent space performed by the information processing device 100. This figure shows an example of predictive control in latent space performed by the information processing device 100. This figure shows an example of the hardware configuration of the information processing device 100.

[0013] Hereinafter, embodiments of the present invention (this embodiment) will be described with reference to the drawings. The embodiments described below are merely examples, and the embodiments to which the present invention is applied are not limited to these embodiments. In this embodiment, the physical quantities targeted are physical quantities of plasma, but the technology according to this embodiment can also be applied to physical quantities other than those of plasma.

[0014] (Regarding the estimation of the equation of state) First, the estimation of the equation of state in this embodiment will be explained in detail. In this embodiment, the information processing device 100, which will be described later, converts plasma physical quantities obtained from plasma measuring instruments into latent variables using a neural network such as AE, and identifies the equation of state using these latent variables as state variables with SINDy. For regularization in the sparse estimation of the equation of state by SINDy, Soothly Clipped Absolute Derivation (SCAD) (Non-Patent Literature 8), a regularization method with oracle properties, is used. This makes it possible to identify the equation of state of a truly sparse system.

[0015] In SINDy, the state variable x(t) ∈ R is a time-series data. ｎ Assuming that the time derivative dx(t) / dt = x'(t) is obtained, and that they obey the following ordinary differential equation, we find the function f by sparse regression.

[0016] Here, x is at each time t １ ,t ２ , , t ｍ If sampling is possible, the matrix representation can be performed as shown in equations (2) and (3) below.

[0017]

[0018] In equations (2) and (3), the rows represent the state index n, and the columns represent the time index m. Rewriting equation (1) as a matrix gives equation (4) below.

[0019] Θ(X) is called a library matrix and is a matrix composed of candidates for non - linear functions of the rows of X. Assuming that Θ(X) consists of constants, polynomials, and trigonometric function terms, Θ(X) is as follows.

[0020] X Ｐ＿２ ,X Ｐ＿３ represents a high - order polynomial. Note that "P_2" intends that 2 is a subscript of P. "X Ｐ＿３ " is the same. For example, X Ｐ＿２ shows the second - order non - linearity in state x as follows.

[0021] Each row of Θ(X) represents a candidate function of Equation (1). Ξ is a matrix consisting of the coefficients of each term of the library matrix. This matrix is composed of the following row vectors ξ ｋ .

[0022] This coefficient vector ξ ｋ is a sparse vector composed of a small number of non - zero elements. That is, under the assumption that the observed information can be represented by a small number of dominant terms, sparse regression is performed in SINDy. When formulating this sparse regression, it can be described as follows.

[0023] In Equation (8), λ is a regularization parameter and p is a regularization term. When the regularization term p is the l １ - norm p(Ξ)=||Ξ|| １ = Σ ｎ ｋ＝１ |ξ ｋ |, it becomes l １ - regularization.

[0024] In this embodiment, as follows, the constraint p(Ξ) is set to SCAD regularization.

[0025] In Equation (9), λ and γ are parameters for adjusting non - convexity. λ ∈ (0, ∞), γ ∈ (1, ∞), and γ = 3.7 is recommended. It is considered that more accurate estimation of the state equation becomes possible by applying oracle - like SCAD regularization.

[0026] Next, for the identification of the low-dimensional model, consider the case where x(t) is converted by the AE into z(t) ∈ R, which is the latent variable in the latent space. The dimension is n > i. Using the expression of the prior research (Non-Patent Document 5), the sparse regression in this case can be expressed as follows. ｉ In this embodiment, the constraint p is replaced with SCAD as in the case of x(t). Considering the architecture configuration including AE and SINDy in the prior research (Non-Patent Document 5), the technology according to this embodiment can be implemented by applying SCAD to the sparse regression part. When only SINDy is considered, it is only necessary to implement the proximity operator of SCAD in the SR3 (Non-Patent Document 9), which is the standard optimization algorithm.

[0027] (Example) Hereinafter, as an example, the system configuration and the operation of the system will be specifically described. Here, consider the model predictive control for the position instability of the core plasma of a tokamak-type fusion reactor. Hereinafter, the tokamak-type fusion reactor is referred to as a tokamak device.

[0028] Fig. 1 shows an overall configuration example of the predictive control system for the plasma in this embodiment. As shown in Fig. 1, this system includes an information processing device 100, a tokamak device 200, a plasma measurement device 300, and a control device 400. The tokamak device 200 is a device that confines ultra-high temperature plasma for nuclear fusion.

[0029] The plasma measurement device 300 is a device that measures the physical quantities of the plasma in the tokamak device 200. The information processing device 100 is a device that performs prediction and control of the plasma based on the physical quantities of the plasma obtained by the plasma measurement device 300. The control device 400 is a device that performs actual control of the tokamak device 200 according to the control instruction (control quantity) received from the information processing device 100.

[0030]

[0031] Figure 2 is a diagram showing the system of Figure 1 in more detail. As shown in Figure 2, the information processing device 100 has a prediction unit 110 (predictive part) and a control unit 120 (control part). The actuator corresponds to the control unit 400. Also in Figure 2, the tokamak device 200 and the plasma measurement device 300 are collectively shown as "plasma". The prediction unit 110 is a functional unit constructed with a neural network such as SINDy and AE. The control unit 120 is a functional unit that calculates optimal control by solving an optimization problem based on an objective function and issues control instructions to the control unit 100.

[0032] The prediction unit 110 and the control unit 120 perform model predictive control based on the measured plasma value m(t) and control u(t), and perform optimal control u(t+1) on the plasma via the control device 400. In this embodiment, the state equation of the prediction part is set to be in latent space, and as described above, an oracle-like SCAD is introduced into the regularization term of its sparse identification.

[0033] Based on the configuration shown in Figure 2, the operation of the information processing device 100 will be explained with reference to the flowchart shown in Figure 3.

[0034] <S1 (Step 1): Input> Measurement values ​​of the plasma in the tokamak device 200, observational information m(t) of the plasma calculated (reconstructed) based on these values ​​(such as the position and shape of the plasma, current distribution, and pressure distribution), and control variables u(t) corresponding to the control by the control device 400 (such as the coil current value and waveform for position control) are input to the prediction unit 110.

[0035] <S2: Conversion to latent variables> In S2, the prediction unit 110 converts the input information x(t) = (m(t)u(t)) ∈ R, which consists of the observed information m(t) and the control variable u(t). ｎ The latent variables z(t) ∈ R in the low-dimensional latent space are extracted by feature extraction using AE or convolutional neural networks. ｉ Convert to i < n.

[0036] For simplicity, we consider a neural network f constructed using only AE, where the transformation to the latent space is expressed as z(t) = f(x(t)). This neural network is a pre-constructed neural network.

[0037] <S3: Identification of State Equation> In S3, the prediction unit 110, constructed using a neural network such as SINDy or AE, identifies the state equation dz(t) / dt = Θ(z(t))Ξ, which is the predictive mathematical model. The method for identifying the state equation dz(t) / dt = Θ(z(t))Ξ has already been explained.

[0038] <S4: Prediction and Control> The control unit 120 calculates the optimal control u(t+1) based on the prediction using the above-mentioned equation of state. The control unit 120 outputs the optimal control u(t+1) to the control device 400 and feeds it back to the plasma that is the target of control. Note that the calculation of the optimal control u(t+1) based on the prediction may be performed by the prediction unit 110.

[0039] <Specific Examples of Model Predictive Control> Figure 4 shows the plasma position change in a two-dimensional latent space (z １ , z ２ The map shows the result when converted to (). The shades of gray in the map represent the amount of plasma position change. Furthermore, since controlling plasma return becomes more difficult the larger the amount of position change, the boundary of the core's stable operating region (marginal boundary) is represented by a black dotted line.

[0040] Furthermore, the solid line represents the trajectory of the time evolution in latent space. In other words, by tracing the latent variables at each time point, the state of the plasma discharge can be represented as a trajectory in latent space. When considering a fusion reactor as a power reactor that requires steady-state operation, it is necessary to prevent this trajectory from exceeding the stability boundary in order to ensure stable operation.

[0041] Figures 5 to 8 ((a) to (d)) show the two-dimensional latent space (z) at the sampling time from time step t to t+1, which is executed by the information processing device 100. １ , z ２This figure shows the process of model predictive control in ).

[0042] Here, we assume that time evolution data up to time t has been obtained. Subsequently, as shown in Figure 5(a), the information processing device 100 uses the state equation identified using SINDy to determine the latent variable z from time t+1 onwards. ｔ＋ｊ We determine the trajectory (i.e., we perform time series forecasting of the latent variable).

[0043] Next, as shown in Figure 6(b), the information processing device 100 controls the target quantity z in the latent space. ｇ The objective function is calculated. In the example in Figure 5(a), the time series prediction results show that the trajectory of the state in the latent space exceeds the stability boundary. Therefore, as shown in Figure 6(b), the control objective is set so that it does not exceed the stability boundary.

[0044] Next, as shown in Figure 7(c), the information processing device 100 searches for the optimal solution (latent action) from the current latent state to the target quantity within the sampling period, and as shown in Figure 8(d), the optimal action (optimal control) u ｔ＋１ Perform the calculation.

[0045] The calculation of optimal control in latent space is obtained by calculating the objective function J and solving its optimization problem, similar to conventional model predictive control. Although there is some arbitrariness in the design of the objective function J, for example, a pre-designed z ｄｅｓｉｇｎ (t) trajectory and z predicted within the sampling period ｐｒｅｄｉｃｔ We can consider the squared error at each time point with respect to the trajectory (t). By solving the optimization problem of J, we can achieve optimal control u ｔ＋１ This can be obtained. Optimal control in latent space u ｔ＋１ Model predictive control is performed by executing corresponding actual control on the plasma through control actuators such as magnetic coils.

[0046] (Example Hardware Configuration) The information processing device 100 described in this embodiment can be realized, for example, by having a computer execute a program. This computer may be a physical computer or a virtual machine on the cloud.

[0047] In other words, the information processing device 100 can be realized by using hardware resources such as the CPU and memory built into the computer to execute a program corresponding to the processing performed by the information processing device 100. The above program can be recorded on a computer-readable recording medium (such as portable memory), saved, and distributed. It is also possible to provide the above program via a network such as the Internet or email.

[0048] Figure 9 shows an example of the hardware configuration of the computer described above. The computer in Figure 9 has a drive device 1000, an auxiliary storage device 1002, a memory device 1003, a CPU 1004, an interface device 1005, a display device 1006, an input device 1007, an output device 1008, etc., all of which are interconnected by bus B. The computer may also be equipped with a GPU.

[0049] The program that enables processing on the computer is provided on a recording medium 1001, such as a CD-ROM or memory card. When the recording medium 1001 containing the program is set in the drive device 1000, the program is installed from the recording medium 1001 to the auxiliary storage device 1002 via the drive device 1000. However, the program does not necessarily have to be installed from the recording medium 1001; it may also be downloaded from another computer via a network. The auxiliary storage device 1002 stores the installed program as well as necessary files and data.

[0050] The memory device 1003 reads and stores a program from the auxiliary storage device 1002 when a program startup command is received. The CPU 1004 implements the functions related to the information processing device 100 according to the program stored in the memory device 1003. The interface device 1005 is used as an interface for connecting to a network, etc. The display device 1006 displays a GUI (Graphical User Interface) etc. based on the program. The input device 1007 consists of a keyboard and mouse, buttons, or a touch panel etc., and is used to input various operation commands. The output device 1008 outputs the calculation results.

[0051] (Summary of Embodiments, Effects, etc.) As described above, in the technology described in this embodiment, the information processing device 100 combines a neural network such as AE with SINDy to obtain a low-dimensional model of the system's state equation, and performs model prediction control of the fusion plasma in latent space based on this prediction. SCAD, a regularization method with oracle properties, is used when identifying sparseness in the state equation. That is, by using SCAD regularization with oracle properties, which is not standardly installed in SINDy, it is possible to identify the exact non-zero elements and their coefficients in the state equation. This makes it possible to construct an accurate predictive mathematical model of the controlled object.

[0052] The technology according to this embodiment is expected to provide the following effects on the control of fusion plasma.

[0053] - By reducing the multidimensional state variable x(t), which is composed of the plasma measurement value m(t) and the control variable u(t), to a lower-dimensional state variable z(t), the degrees of freedom in the problem can be reduced.

[0054] Identifying the predictive model as a state equation makes it possible to provide interpretability, generalization, and extrapolability.

[0055] - It becomes possible to construct low-dimensional models by combining AE with other feature-extracting architectures, such as convolutional neural networks.

[0056] • By applying SCAD, the state equation can be estimated more accurately, which can contribute to the realization of model predictive control where accurate predictive models are required.

[0057] While nuclear fusion plasma was cited as an example of a controlled object, the technology according to the present invention can be used for other control problems where model predictive control is employed, such as in automated driving and plant control.

[0058] The following additional information is disclosed regarding the embodiments described above.

[0059] <Notes> (Note 1) An information processing device comprising: a prediction unit that calculates an objective function by converting input information consisting of observational information of a predetermined physical quantity and control variables for the predetermined physical quantity into latent variables in a low-dimensional latent space, identifying a state equation by sparse regression with oracle-like regularization applied using the latent variables, and performing time series prediction using the state equation; and a control unit that calculates optimal control by solving an optimization problem based on the objective function. (Note 2) The information processing device according to Note 1, wherein the predetermined physical quantity is a physical quantity of plasma in a tokamak-type nuclear fusion reactor. (Note 3) An information processing method executed by an information processing device, comprising the steps of: converting input information consisting of observational information of a predetermined physical quantity and control variables for the predetermined physical quantity into latent variables in a low-dimensional latent space, identifying a state equation by sparse regression with oracle-like regularization applied using the latent variables, and calculating an objective function by performing time series prediction using the state equation; and calculating optimal control by solving an optimization problem based on the objective function. (Appendix 4) A program for causing a computer to function as a component of the information processing device described in Appendix 1 or 2.

[0060] Although this embodiment has been described above, the present invention is not limited to this specific embodiment, and various modifications and changes are possible within the scope of the gist of the invention as described in the claims.

[0061] 100 Information processing device 110 Prediction unit 120 Control unit 200 Tokamak device 300 Plasma measuring device 400 Control device 1000 Drive device 1001 Recording medium 1002 Auxiliary storage device 1003 Memory device 1004 CPU 1005 Interface device 1006 Display device 1007 Input device 1008 Output device

Claims

1. An information processing device comprising: a prediction unit that converts input information consisting of observational information of a predetermined physical quantity and control variables for the predetermined physical quantity into latent variables in a low-dimensional latent space, identifies a state equation by sparse regression with oracle-like regularization applied using the latent variables, and calculates an objective function by performing time series prediction using the state equation; and a control unit that calculates optimal control by solving an optimization problem based on the objective function.

2. The information processing apparatus according to claim 1, wherein the predetermined physical quantity is a physical quantity of plasma in a tokamak-type nuclear fusion reactor.

3. An information processing method executed by an information processing device, comprising the steps of: converting input information consisting of observational information of a predetermined physical quantity and control variables for the predetermined physical quantity into latent variables in a low-dimensional latent space; identifying a state equation by sparse regression with oracle-like regularization applied using the latent variables; and calculating an objective function by performing time series forecasting using the state equation; and calculating optimal control by solving an optimization problem based on the objective function.

4. A program for causing a computer to function as a component of the information processing apparatus described in claim 1 or 2.

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