Physical field prediction model learning device, physical field prediction model learning method, and physical field prediction model learning program

Abstract

To enhance prediction accuracy of a physical field.SOLUTION: A learning device 2 includes: an input unit 21 which accepts teacher data D4 including learning data D1 indicating an initial state of physical fields X, Y, and P, and correct answer data D3 obtained by applying the learning data D1 to a dominant equation E dominating the physical fields X, Y, and P; a prediction unit 22 which obtains a prediction value for a physical quantity defined for each of plurality of nodes N by applying the learning data D1 to a learning model M for predicting the physical fields X, Y, and P; an evaluation unit 23 which evaluates the prediction value of the physical quantity by using the correct answer data D3; and an adjustment unit 24 which adjusts a parameter of the learning model M by receiving a result from the evaluation unit 23. The evaluation unit 23 evaluates the prediction value of the physical quantity by using a loss function L. The loss function L includes a term regarding a gradient expressed by a prediction value of a physical quantity defined by a first node N and a prediction value defined by a second node N different from the first node.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] The present invention relates to a physical field prediction model learning device, a physical field prediction model learning method, and a physical field prediction model learning program. [Background technology]

[0002] In the development of industrial products, it is common to construct a virtual model of the target product on a computer and conduct virtual tests on the computer. For example, the results of a virtual test can be obtained as a numerical solution to a governing equation given several conditions through numerical calculation. In recent years, methods for conducting similar virtual tests using machine learning techniques have been investigated. For example, Patent Documents 1 to 3 disclose a method for predicting the physical field using machine learning instead of numerically solving the governing equation. [Prior art documents] [Patent Documents]

[0003] [Patent Document 1] Japanese Patent Publication No. 2020-191096 [Patent Document 2] Japanese Patent Publication No. 2021-135775 [Patent Document 3] Japanese Patent Publication No. 2021-157682 [Overview of the Initiative] [Problems that the invention aims to solve]

[0004] Supervised learning is sometimes used to obtain a learning model for predicting physical fields. In supervised learning, the learning device is provided with problem data (training data) and correct answer data for the problem (ground truth data). The learning device then updates its learning model so that the prediction results, based on the training data, approach the ground truth data.

[0005] In this case, the correct data is represented by a space that is composed of multiple nodes, each defined by a physical quantity. In other words, the learning model predicts the physical quantities for each node contained in the space. While such a learning model can predict the physical quantities for each node, it cannot necessarily predict with high accuracy the relationship between the physical quantities defined for one node and those defined for another.

[0006] This invention provides a physical field prediction model learning device, a physical field prediction model learning method, and a physical field prediction model learning program that can improve the accuracy of physical field prediction. [Means for solving the problem]

[0007] A physical field prediction model learning device, one embodiment of the present invention, is a physical field prediction model learning device for predicting a physical field composed of physical quantities defined for each of the multiple nodes in a region represented by a plurality of nodes, and comprises: an input unit that accepts training data including training data indicating the initial state of the physical field and ground truth data obtained as a result of applying the training data to governing equations that govern the physical field; a prediction unit that obtains predicted values ​​of physical quantities defined for each of the multiple nodes by applying the training data to a learning model for predicting the physical field; an evaluation unit that evaluates the predicted values ​​of physical quantities using the ground truth data; and an adjustment unit that adjusts the parameters of the learning model based on the results of the evaluation unit. The evaluation unit evaluates the predicted values ​​of physical quantities using a loss function. The loss function includes a term relating to the gradient expressed by the predicted value of a physical quantity defined for a first node and the predicted value of a physical quantity defined for a second node different from the first node.

[0008] The physical field prediction model learning device uses a loss function that includes a term representing the gradient. Evaluation using such a loss function allows for accurate prediction of not only the physical quantities obtained at each node, but also the relationships (gradients) between those quantities, enabling adjustment of the learning model's parameters. As a result, the accuracy of physical field prediction can be improved.

[0009] The loss function in the above physical field prediction model learning device includes a first term, a second term, and a third term. The first term includes the difference between the true value of the physical quantity given as the correct answer data and the predicted value of the physical quantity output by the prediction unit. The second term includes a governing equation including an element for differentiating the physical quantity as a term related to the gradient. The third term may include a boundary condition given as a Dirichlet boundary condition. By evaluating such a loss function, the parameters of the learning model can also be adjusted so that the gradient of the physical quantity can be accurately predicted.

[0010] The loss function in the above physical field prediction model learning device includes a first term, a second term, and a third term. The first term includes the difference between the true value of the physical quantity given as the correct answer data and the predicted value of the physical quantity output by the prediction unit. The second term includes a governing equation that does not include an element for differentiating the physical quantity as a term related to the gradient. The third term may include a boundary condition given as a Neumann boundary condition as a term related to the gradient. By evaluating such a loss function, the parameters of the learning model can also be adjusted so that the gradient of the physical quantity can be accurately predicted.

[0011] The loss function in the above physical field prediction model learning device includes at least a first term, and the first term may include the true value related to the gradient of the physical quantity and the predicted value related to the gradient of the physical quantity. By evaluating such a loss function, the parameters of the learning model can also be adjusted so that the gradient of the physical quantity can be accurately predicted.

[0012] The loss function in the above physical field prediction model learning device may include an operation of weighted averaging the first term, the second term, and the third term. By evaluating such a loss function, the parameters of the learning model can also be adjusted so that the gradient of the physical quantity can be accurately predicted.

[0013] Another embodiment of the present invention is a physical field prediction model learning method for predicting a physical field composed of physical quantities defined for each of the multiple nodes in a region represented by a plurality of nodes, comprising the steps of: receiving training data including training data representing the initial state of the physical field and ground truth data obtained as a result of applying the training data to governing equations that govern the physical field; obtaining predicted values ​​of physical quantities defined for each of the multiple nodes by applying the training data to a learning model for predicting the physical field; evaluating the predicted values ​​of physical quantities using the ground truth data; and adjusting the parameters of the learning model based on the results of the evaluation step. The evaluation step evaluates the predicted values ​​of physical quantities using a loss function. The loss function includes a term relating to the gradient expressed by the predicted value of a physical quantity defined for a first node and the predicted value of a physical quantity defined for a second node adjacent to the first node.

[0014] A further embodiment of the present invention is a physical field prediction model learning program for predicting a physical field composed of physical quantities defined for each of the multiple nodes in a region represented by multiple nodes, and causes a computer to perform the following steps: receiving training data including training data representing the initial state of the physical field and ground truth data obtained as a result of applying the training data to governing equations that govern the physical field; applying the training data to a learning model for predicting the physical field to obtain predicted values ​​of the physical quantities defined for each of the multiple nodes; evaluating the predicted values ​​of the physical quantities using the ground truth data; and adjusting the parameters of the learning model based on the results of the evaluation step. The evaluation step evaluates the predicted values ​​of the physical quantities using a loss function. The loss function includes a term relating to the gradient expressed by the predicted value of the physical quantity defined for the first node and the predicted value of the physical quantity defined for the second node adjacent to the first node.

[0015] The accuracy of physical field predictions can also be improved through physical field prediction model learning methods and physical field prediction model learning programs. [Effects of the Invention]

[0016] According to the present invention, a physical field prediction model learning device, a physical field prediction model learning method, and a physical field prediction model learning program are provided that can improve the accuracy of predicting physical fields. [Brief explanation of the drawing]

[0017] [Figure 1] Figure 1 is a functional block diagram of the physical field prediction system, including the physical field prediction model learning device. [Figure 2] Figure 2(a) is a conceptual diagram illustrating a physical field prediction device. Figure 2(b) is a conceptual diagram illustrating a numerical analysis device. [Figure 3] Figure 3 shows an example of the data defined for the physical field prediction model learning device. [Figure 4] Figures 4(a), 4(b), and 4(c) show examples of numerical calculation methods for obtaining gradients. [Figure 5] Figure 5 shows the configuration diagram of the computer that realizes the physical field prediction device. [Figure 6] Figure 6 is a flowchart of the physical field prediction method. [Figure 7] Figures 7(a) and 7(b) illustrate the effects of the physical field prediction device. [Figure 8] Figures 8(a) and 8(c) show the prediction results of the physical field using the comparative example's learning model. Figures 8(b) and 8(d) show the prediction results of the physical field using the example's learning model. [Figure 9] Figure 9 is a graph used to evaluate the prediction results using the learning model of the example. [Modes for carrying out the invention]

[0018] The physical field prediction model learning apparatus, physical field prediction model learning method, and physical field prediction model learning program according to the embodiment will be described in detail below with reference to the attached drawings. In the description of the drawings, the same elements are denoted by the same reference numerals, and redundant explanations are omitted.

[0019] Figure 1 is a functional block diagram illustrating the physical field prediction device 1. The physical field prediction device 1 predicts the physical field using a physical field prediction model. The physical field prediction model is a learning model M generated by the physical field prediction model learning device (hereinafter referred to as learning device 2) included in the physical field prediction device 1.

[0020] In this embodiment, the physical fields X, Y, and P are defined as a two-dimensional or three-dimensional region defined by a plurality of nodes N, as shown in Figure 2(a), and are composed of physical quantities defined for each of these nodes N.

[0021] For example, a two-dimensional prediction region is divided into multiple nodes N. Dividing a prediction region into multiple nodes N is also called meshing. Each node N in the prediction region is assigned, for example, a velocity in the X direction. In other words, the prediction region can be said to be a physical field X for predicting the velocity in the X direction. If a velocity in the Y direction is defined for node N, then the prediction region can be said to be a physical field Y for predicting the velocity in the Y direction. If pressure is defined for node N, then the prediction region can be said to be a physical field P for predicting pressure.

[0022] For example, the physical field prediction device 1 receives problem data DP, which is a set of initial conditions defined for node N in the prediction region. These initial conditions include the velocity in the X direction, the velocity in the Y direction, and the pressure. In addition, boundary conditions are given to the boundary node BN that defines the boundary of the prediction region. For example, at boundary node BN, the values ​​of the velocity in the X direction and the velocity in the Y direction may be set to zero as boundary conditions to indicate that no flow is occurring.

[0023] Next, the physical field prediction device 1 applies the initial conditions and boundary conditions to the learned model M. As a result, the physical field prediction device 1 obtains, for example, the physical quantities at each node N after a desired time has elapsed (t=n), using the time at which the state indicated by the initial conditions is defined as the reference time (t=0), as predicted data D2.

[0024] This operation, focusing only on input information (initial conditions, boundary conditions) and output information (predicted physical field), is similar to a computer simulation using a so-called numerical analysis device 4, as shown in Figure 2(b). In general computer simulations, the analysis domain is divided (meshed) using a preprocessor 41. Next, after setting initial conditions and boundary conditions for the mesh, the simulation is executed using a solver 42. Here, the calculation performed by the solver 42 is to solve the governing equation E, to which the initial conditions and boundary conditions are applied, by numerical analysis. In other words, the numerical analysis device 4 performs the process of solving the governing equation E by numerical analysis in order to obtain output information from the input information. Then, the results output by the solver 42 are visualized using a postprocessor 43.

[0025] In contrast, the physical field prediction device 1 does not perform the process of numerically solving the governing equation E in order to obtain output information from input information. The physical field prediction device 1 obtains output information by applying the input information to the learning model M.

[0026] Generally, numerically solving the governing equation E is computationally intensive. In other words, the time required to obtain one output piece of information corresponding to one input piece tends to be long. On the other hand, processing using a learning model M tends to be less computationally intensive than numerically solving the governing equation E. In other words, the time required to obtain one output piece of information corresponding to one input piece can be shorter than in the above case. In product development using computer simulations, feedback is often repeated, such as setting input information, obtaining output information, and then modifying the input information based on the results. Since the physical field prediction device 1 has a low computational load from input information to output information, this feedback can be repeated multiple times within a given time. As a result, it becomes possible to obtain design results with higher accuracy.

[0027] The following provides a detailed description of the physical field prediction device 1.

[0028] As shown in Figure 1, the physical field prediction device 1 includes several functional components. These functional components are realized by the computer 500, described later, executing the physical field prediction model learning program PG.

[0029] The physical field prediction device 1 comprises a learning device 2 and an inference device 3. Although a numerical analysis device 4 is shown in Figure 1, the numerical analysis device 4 is not included as a component of the physical field prediction device 1.

[0030] Numerical analysis device 4 receives training data D1 and outputs correct data D3. Numerical analysis device 4 solves the governing equation E, to which the initial conditions shown in training data D1 are applied, by numerical analysis. For example, when physical field prediction device 1 predicts a physical field that is a flow field, the equation governing the flow field is the Navier-Stokes equation. Therefore, numerical analysis device 4 also solves the Navier-Stokes equation, to which initial conditions including initial velocity and initial pressure are set, by numerical analysis.

[0031] Learning device 2 generates a learning model M for predicting the physical fields X, Y, and P. Learning device 2 uses training data D4 to learn how to obtain the learning model M. In other words, learning device 2 performs so-called supervised learning. Inference device 3 predicts the physical fields X, Y, and P using the learning model M. Inference device 3 uses training data D1 and the learning model M to predict the physical fields X, Y, and P.

[0032] The learning device 2 includes an input unit 21, a prediction unit 22, an evaluation unit 23, an adjustment unit 24, and an output unit 25.

[0033] The input unit 21 receives training data D4 for learning. The training data D4 includes training data D1 corresponding to the input information used in the inference process of the inference device 3, and ground truth data D3, which is the result that the inference device 3 should aim to predict. The training data D1 may be pre-set by a user using the physical field prediction device 1. The ground truth data D3 may be obtained by inputting the training data D1 into the numerical analysis device 4. In other words, the ground truth data D3 is obtained by general computer simulation. The ground truth data D3, like the training data D1, is composed of multiple nodes N that define the prediction region and physical quantities defined for each node N.

[0034] The prediction unit 22 inputs the training data D1 into the learning model M to obtain predicted data D2, which is the result of predicting the physical fields X, Y, and P.

[0035] The evaluation unit 23 evaluates the physical fields X, Y, and P predicted by the prediction unit 22. The evaluation is obtained as the value of the loss function L.

[0036] Equation (1) shows the loss function L.

number

[0037] The right side of Equation (1) includes the absolute error between the predicted value and the true value (the first term), the error when the predicted value is substituted into the governing equation E (the second term), and the error when the predicted value is substituted into the boundary conditions (the third term). The evaluation unit 23 multiplies each of these three terms by weights (ω L1 , ω NS , ω BC ). Then, the evaluation unit 23 adds up the results multiplied by the weights (ω L1 , ω NS , ω BC ). That is, the evaluation unit 23 performs weighted average processing on the three terms.

[0038] The first term on the right side of Equation (1) evaluates the difference between the true value of the physical quantity corresponding to node N and the predicted value of the physical quantity corresponding to node N. The variable (ω L1 ) is a weight. The variable (L L1 ) is a value related to the difference between the true value and the predicted value. The variable (L L1 ) is as shown in Equation (2).

Equation

[0039] Note that the first term of the loss function L described above used the absolute error. As the first term of the loss function L, instead of the absolute error, the squared error or the evaluation value of SSIM (Structural Similarity) described later may be used. Also, the first term of the loss function L may be a combination of the absolute error, the squared error, and the evaluation value of SSIM.

[0040] The second term on the right-hand side of equation (1) evaluates the value obtained when the predicted value corresponding to node N is substituted into the governing equation E. L2 ) is the weight. Variable (L NS ) is the value obtained by substituting the predicted value corresponding to node N into the governing equation E. NS The result is as shown in equation (3).

number

[0041] Here, f NS This is the governing equation E that runs through the physical field. For example, if the predicted region is a flow field, the equations governing the flow field are the Navier-Stokes equations shown in equation (4).

number

number

[0042] Here, equation (5) includes an element that differentiates a physical quantity. Specifically, equation (5) includes a spatial derivative term (∇). Since the spatial derivative term (∇) represents a gradient, it can also be said that equation (5) includes a term corresponding to the velocity gradient and a term corresponding to the pressure gradient. In other words, the fact that the loss function L includes equation (5) which includes the spatial derivative term (∇) means that the loss function L includes a term corresponding to the velocity gradient and a term corresponding to the pressure gradient. Therefore, the loss function (L: equation (1)) does not simply evaluate the predicted value based on the difference between the true value and the predicted value for each node N. In addition to the difference between the true value and the predicted value for each node N, the loss function (L: equation (1)) also evaluates the gradient of the values ​​defined for each node N (velocity gradient and pressure gradient).

[0043] Note that the element used to differentiate a physical quantity was the spatial derivative term. However, the element used to differentiate a physical quantity is not limited to the spatial derivative term. For example, the element used to differentiate a physical quantity may be the distance derivative term or the time derivative term.

[0044] Adjusting the parameters of the learning model M based on the difference between the true value and the predicted value for each node N can be considered local fitting. In contrast, adjusting the parameters of the learning model M based on the gradient between the value defined for a certain node N and the value defined for another node N adjacent to that node N can be considered global fitting. Therefore, the evaluation unit 23 performs both local and global fitting.

[0045] Referring to Figure 4, an example of how to calculate spatial derivatives will be explained. The spatial derivative term is calculated using the finite difference method. In this example, an example of calculating velocity in the X direction is shown. The gradient can be calculated for velocity in the Y direction and pressure using similar calculations. For example, as shown in region A1 of Figures 4(a) and 4(b), if the boundary node BN is not included, the gradient can be obtained by equations (6) and (7). The differential calculations shown in equations (6) and (7) are second-order accurate central differences.

number

number

[0046] For example, if the region A2 in Figures 4(a) and 4(c) includes the boundary node BN, the gradient can be obtained by equations (8) and (9). The differential calculations shown in equations (8) and (9) are second-order precision one-sided differences.

number

number

[0047] The third term on the right-hand side of equation (1) evaluates the value obtained when the predicted value of the physical quantity corresponding to the boundary node BN is substituted into the boundary conditions. Variable (ω L3 ) is the weight. Variable (L BC ) is the value obtained when the predicted value of the physical quantity is substituted into the boundary conditions. Variable (L BC The following is shown in equation (10). The boundary conditions are as shown in equation (11). The boundary conditions shown in equation (11) are predetermined numerical values ​​defined for each node N. In other words, the boundary conditions do not contain any information about the gradient. Boundary conditions that do not contain any information about the gradient are also called Dirichlet boundary conditions.

number

number

[0048] As stated, the loss function L (Equation (1)) is weighted averaged. When summing the absolute error between the predicted value and the true value (1st term), the error when the predicted value is substituted into the governing equation (2nd term), and the error when the predicted value is substituted into the boundary conditions (3rd term), the weight (ω) is the factor that determines how much importance each term is given. L1 , ω NS , ω BC ) is. For example, weight (ω L1 , ω NS , ω BC If we set all of ) to 1, all terms will be considered equally, weight (ω L1 , ω NS , ω BC The result is shown in equations (12) to (15).

number

number

number

number

[0049] The adjustment unit 24 adjusts the parameters of the learning model M using the results from the evaluation unit 23. The adjusted learning model M is then used again in the prediction unit 22 to predict the physical fields X, Y, and P.

[0050] The output unit 25 outputs the learning model M to the inference device 3 using the results from the evaluation unit 23. For example, if the value of the loss function L output by the evaluation unit 23 satisfies a predetermined condition, it can be determined that learning has finished. In that case, the output unit 25 outputs the learning model M, which has finished learning, to the inference device 3.

[0051] The inference device 3 receives the training data D1. The inference device 3 applies the training data D1 to the training model M to obtain predictions for the physical fields X, Y, and P. Specifically, the inference device 3 calculates physical quantities such as velocity in the X direction, velocity in the Y direction, or pressure for each node N.

[0052] Referring to Figure 5, the hardware configuration of the physical field prediction device 1 will be described. The physical field prediction device 1 is realized, for example, by the execution of a physical field prediction model learning program PG by a computer 500. The physical field prediction model learning program PG causes the computer 500 to execute a physical field prediction model learning method.

[0053] The physical field prediction device 1 includes one or more computers 500. The computer 500 has a processor, a CPU (Central Processing Unit) 501, a main memory unit 502, an auxiliary memory unit 503, a communication control unit 504, an input device 505, and an output device 506. The physical field prediction device 1 is composed of one or more computers 500, which consist of this hardware and software such as programs.

[0054] If the physical field prediction device 1 is composed of multiple computers 500, these computers 500 may be connected locally or via a communication network such as the Internet or an intranet. This connection logically constructs a single physical field prediction device 1.

[0055] The CPU 501 executes the operating system and application programs. The main memory 502 consists of ROM (Read Only Memory) and RAM (Random Access Memory). The auxiliary memory 503 is a storage medium consisting of a hard disk and flash memory. The auxiliary memory 503 generally stores a larger amount of data than the main memory 502. The communication control unit 504 consists of a network card or wireless communication module. The input device 505 consists of a keyboard, mouse, touch panel, and microphone for voice input. The output device 506 consists of a display and printer.

[0056] The auxiliary storage unit 503 stores the physical field prediction model learning program PG and the data necessary for processing in advance. The physical field prediction model learning program PG causes the computer 500 to execute each functional element of the physical field prediction device 1. For example, the physical field prediction model learning program PG is read by the CPU 501 or the main memory unit 502 and operates at least one of the CPU 501, main memory unit 502, auxiliary storage unit 503, communication control unit 504, input device 505, and output device 506. For example, the physical field prediction model learning program PG reads and writes data to the main memory unit 502 and the auxiliary storage unit 503.

[0057] The physical field prediction model learning program PG may be provided on a tangible recording medium such as a CD-ROM, DVD-ROM, or semiconductor memory. Alternatively, the physical field prediction model learning program PG may be provided as a data signal via a communication network.

[0058] Next, we will explain the physical field prediction method, including the physical field prediction model learning method, with reference to the flowchart shown in Figure 6.

[0059] The physical field prediction method includes a step of preparing training data D4 (step S10), a learning step (step S20), and an inference step (step S30). The learning step (step S20) corresponds to the physical field prediction model learning method.

[0060] In the step of preparing the training data D4 (step S10), first, the learning data D1 is prepared (step S11). The learning data D1 may be input by a user using the physical field prediction device 1 using the input device 505. Alternatively, the learning data D1 may be read from an external storage device. Alternatively, the learning data D1 may be received from an external source via the communication control unit 504. Next, the correct answer data D3 is prepared (step S12). Step S12 may be performed by the numerical analysis device 4. Also, similar to the learning data D1, it may be read from an external storage device. Alternatively, the correct answer data D3 may be received from an external source via the communication control unit 504. Through the steps S11 and S12 described above, the learning data D1 and the correct answer data D3 are obtained. In other words, the training data D4 is obtained.

[0061] In the learning step (step S20) of the physical field prediction model learning method, first, training data D4 is received (step S21). Step S21 is performed by the input unit 21. Next, prediction data D2 is obtained (step S22). Step S22 is performed by the prediction unit 22. Next, the loss function L is calculated (step S23). Step S23 is performed by the evaluation unit 23. Next, it is determined whether the loss function L satisfies the conditions (step S24). If the loss function L does not satisfy the predetermined conditions (step S24: NO), it is determined that learning will not be terminated. As a result, the parameters of the learning model M are adjusted by the adjustment unit 24. Then, the process from step S22 is repeated. On the other hand, if the loss function L satisfies the predetermined conditions (step S24: YES), it is determined that learning will be terminated. As a result, the learning model M is output to the inference device 3 as a trained model by the output unit 25 (step S26).

[0062] In the inference step (step S30), first, the problem data DP is received (step S31). Next, the prediction data D2 is obtained (step S32). Step S32 is performed by the inference device 3. Then, the obtained prediction data D2 is output (step S33).

[0063] <Effects and Effects> Before explaining the effects of the physical field prediction model learning device of this embodiment, we will provide two reference results that may occur when the loss function includes only the absolute error term, as comparative examples.

[0064] Figure 7(a) shows the first reference results. Points labeled (TV1, TV2, TV3) indicate the true values ​​at nodes N1, N2, and N3, respectively. Points labeled (PR1, PR2, PR3) indicate the predicted values ​​at nodes N1, N2, and N3, respectively.

[0065] First, let's focus on the absolute value of the difference between the physical quantities given to each node N1, N2, and N3. The predicted value PR1 for the first node N1 is greater than the true value TV1 by Δx. The predicted value PR2 for the second node N2 is greater than the true value TV2 by Δx. The predicted value PR3 for the third node N3 is greater than the true value TV3 by Δx. When these differences are expressed as the average of the absolute values ​​of the differences, we get Δx.

[0066] Next, let's look at the gradient between nodes N1 and N2. In the first reference result, the gradient between the first node N1 and the second node N2 has an upward slope for the true values ​​TV1 and TV2. For the predicted values ​​PR1 and PR2, the predicted value PR1 for the first node N1 is Δx greater than the true value, and the predicted value PR2 for the second node N2 is Δx greater than the true value. In other words, the gradient between the predicted values ​​PR1 and PR2 is the same as the gradient between the true values ​​TV1 and TV2 in that it is upward sloping.

[0067] Figure 7(b) shows the second set of reference results. The predicted value PR1 for the first node N1 is greater than the true value TV1 by Δx. The predicted value PR2 for the second node N2 is less than the true value TV2 by Δx. The predicted value PR3 for the third node N3 is greater than the true value TV3 by Δx. When these differences are expressed as the average of the absolute values ​​of the differences, it becomes Δx.

[0068] Next, let's look at the gradient between nodes N1 and N2. In the second reference result, the gradient between the first node N1 and the second node N2 is upward sloping for the true values ​​TV1 and TV2. On the other hand, for the predicted values ​​PR1 and PR2, the predicted value PR1 of the first node N1 is greater than the true value TV1 by Δx, and the predicted value PR2 of the second node N2 is less than the true value TV2 by Δx. In this case, the gradient between the predicted values ​​PR1 and PR2 is zero.

[0069] In other words, when looking at the average of the absolute values ​​of the differences, both the first and second reference results are evaluated as being the same. However, when looking at the gradient, in the first reference result, the gradient between the predicted values ​​PR1 and PR2 is upward sloping, which is the same as the gradient between the true values ​​TV1 and TV2. However, in the second reference result, while the gradient between the true values ​​TV1 and TV2 is upward sloping, the gradient between the predicted values ​​PR1 and PR2 is zero. In other words, in the second reference result, the predicted values ​​PR1, PR2, and PR3 cannot be said to have accurately predicted the gradient. Furthermore, if the governing equation E includes a spatial derivative term, the predicted values ​​cannot be said to satisfy the governing equation E.

[0070] Therefore, the learning device 2, which is a physical field prediction model learning device of this embodiment, predicts physical fields X, Y, and P composed of physical quantities defined for each of the multiple nodes N in a region represented by multiple nodes N. The learning device 2 includes an input unit 21 that accepts training data D4, which includes training data D1 showing the initial state of the physical fields X, Y, and P, and ground truth data D3 obtained as a result of applying the training data D1 to the governing equations E that govern the physical fields X, Y, and P; a prediction unit 22 that obtains predicted values ​​of physical quantities defined for each of the multiple nodes N by applying the training data D1 to a learning model M for predicting the physical fields X, Y, and P; an evaluation unit 23 that evaluates the predicted values ​​of physical quantities using the ground truth data D3; and an adjustment unit 24 that adjusts the parameters of the learning model M based on the results of the evaluation unit 23. The evaluation unit 23 evaluates the predicted values ​​of physical quantities using a loss function L. The loss function L includes a term relating to the gradient expressed by the predicted values ​​of physical quantities defined in a first node N and the predicted values ​​of physical quantities defined in a second node N different from the first node N.

[0071] The physical field prediction model learning device 2 of this embodiment uses a loss function L that includes a gradient term in the generation of the learning model M. Specifically, the loss function L includes the governing equation E (Navier-Stokes equations) which includes a spatial derivative term as a term that indicates the gradient of velocity and pressure. By using evaluation with such a loss function L, it is possible to adjust the parameters of the learning model M so that not only can the velocity and pressure obtained for each node N be predicted with high accuracy, but the relationships (gradients) between the velocities and pressures obtained for each node N can also be predicted with high accuracy. Consequently, the physical field prediction model learning device 2 can obtain predicted values ​​of velocity and pressure that satisfy the governing equation E.

[0072] The loss function L includes the first term shown in equation (2), the second term shown in equation (3), and the third term shown in equation (10). The first term includes the difference between the true value of the physical quantity given as ground truth data and the predicted value of the physical quantity output by the prediction unit. The second term includes the governing equations, which include the spatial derivative of the physical quantity as a gradient term. The third term includes the boundary conditions given as Dirichlet boundary conditions. By evaluation using such a loss function L, the parameters of the learning model M can be adjusted so that the gradient of the physical quantity can be predicted with high accuracy.

[0073] The loss function L includes a weighted average operation of the first term shown in equation (2), the second term shown in equation (3), and the third term shown in equation (10). By evaluating using such a loss function L, the parameters of the learning model M can be adjusted so that the gradient of the physical quantity can be predicted with high accuracy.

[0074] The physical field prediction model learning device, physical field prediction model learning method, and physical field prediction model learning program of the present invention are not limited to the embodiments described above.

[0075] The loss function L in the embodiment included the governing equations, which include spatial derivative terms, as elements. The loss function L may also include a gradient term as an element separate from the governing equations E.

[0076] <Example 1> For example, boundary conditions may include elements related to gradients. Such boundary conditions are called so-called von Neumann boundary conditions. When boundary conditions include elements related to gradients, the governing equation E may or may not include a spatial derivative term. In short, the loss function in Modification 1 includes the first term, the second term, and the third term. The first term includes the difference between the true value of the physical quantity given as ground truth data and the predicted value of the physical quantity output by the prediction unit. The second term may be a governing equation that does not include the distance derivative or spatial derivative of the physical quantity as a gradient term. The third term includes the boundary conditions given as von Neumann boundary conditions as a gradient term. The parameters of the learning model M can also be adjusted so that the gradient of the physical quantity can be predicted with high accuracy by evaluating such a loss function.

[0077] <Modification 2> For example, the training data D4 may include, as the ground truth data D3, the physical quantities defined for each node N, as well as the values ​​of the gradients that are considered to be correct. In short, the loss function in Modification 2 includes at least one term. The first term includes the true value of the gradient of the physical quantity and the predicted value of the gradient of the physical quantity. By evaluating such a loss function, the parameters of the learning model M can be adjusted so that the gradient of the physical quantity can be predicted with good accuracy.

[0078] Below, we present two examples of evaluation results obtained for the physical field predicted by the physical field prediction device 1 of this embodiment.

[0079] <Example 1> In Example 1, the physical field P relating to pressure was predicted using the physical field prediction device 1 of the embodiment. In addition, in Example 1, the physical field P relating to pressure was predicted using a learning model obtained with a loss function that does not include a gradient term, as a comparison. In other words, in Example 1, a comparison was made between a prediction using a learning model obtained with a loss function L that includes a gradient term (embodiment) and a prediction using a learning model obtained with a loss function that does not include a gradient term (comparative example).

[0080] The main conditions for the calculation are as follows: • The same training data was used to perform 10 training sessions. The number of parameter updates for the learning model was standardized to 100,000. Reynolds number Re = 2000. • Weight L L1 :L NS =3:1.

[0081] Figures 8(a) to 8(d) show the results of evaluating how similar contour plots of ground truth data and predicted values ​​are. Figures 8(a) and 8(b) show the results of evaluation using MAPE (Mean Absolute Precentage Error). Figures 8(c) and 8(d) show the results of evaluation using SSIM (Structural Similarity).

[0082] The MAPE shown in equation (16) is the error rate of the predicted data relative to the ground truth data. A smaller error rate indicates that the predicted data is closer to the ground truth data. Furthermore, in the contour plots of Figures 8(a) and 8(b), areas closer to white indicate that the MAPE value is close to zero, while areas closer to black indicate that the MAPE value is close to 100.

number

[0083] The SSIM shown in equation (17) evaluates the correlation around a pixel by obtaining the mean, variance, and covariance of the pixels contained within the set window. The maximum value of the SSIM is 1, and the minimum value is 0. The closer the SSIM value is to 1, the more similar the two images are. In this example, it indicates that the contour plot of the predicted data is similar to the contour plot of the ground truth data. In other words, it indicates high prediction accuracy. Furthermore, in the contour plots of Figures 8(c) and 8(d), areas that are close to white indicate that the SSIM value is close to 1, and areas that are close to black indicate that the SSIM value is close to 0.

number

[0084] Referring to Figure 8(a), which shows the results for the comparative example, it can be seen that gray to black areas exist irregularly. In other words, it can be seen that the MAPE values ​​are distributed between approximately 40 and 100. Referring to Figure 8(c), it can be seen that gray to black areas exist irregularly. In other words, it can be seen that the SSIM values ​​are distributed between approximately 0 and 0.6.

[0085] Referring to Figure 8(b), which shows the results for Example 1, it can be seen that the white areas make up the majority of the area, excluding the corner regions. In other words, it can be seen that the majority of the area has MAPE values ​​between 0 and 20. Referring to Figure 8(d), it can be seen that the white areas make up the entire area. In other words, it can be seen that the SSIM values ​​are distributed between approximately 0.9 and 1.0.

[0086] In other words, a comparison between predictions made by a learning model M obtained using a loss function L that includes a gradient term (embodiment) and predictions made by a learning model obtained using a loss function that does not include a gradient term (comparative example) revealed that predictions made by the learning model M obtained using a loss function L that includes a gradient term yield predicted values ​​that are close to the ground truth data, as well as gradients based on the predicted values.

[0087] <Example 2> In Example 2, we examined the effect of weights. More specifically, the loss function L performs a weighted average using weights. We investigated whether the degree to which the term that evaluates the absolute error (equation (2)) and the term that includes the gradient (equation (3)) are considered (the magnitude of the weights) in this weighted average affects the prediction results.

[0088] The main conditions for the calculation are as follows: • The same training data was used to perform 10 training sessions. The number of parameter updates for the learning model was standardized to 100,000. Reynolds number: Re=2000.

[0089] Furthermore, the following three conditions were set to evaluate the effect of the weights. ·First condition L L1 :L NS = 1:1. ·Second condition L L1 :L NS =3:1. ·Third condition L L1 :L NS = 10:1. The first condition means that the term evaluating the absolute error and the term including the gradient are given equal weight. The second condition means that the term evaluating the absolute error is given three times more weight than the term including the gradient. The third condition means that the term evaluating the absolute error is given ten times more weight than the term including the gradient.

[0090] Figure 9 is a graph showing the results of Example 2. The horizontal axis represents the number of training data points. The vertical axis represents the SSIM value. In other words, the closer the value on the vertical axis is to 1, the higher the prediction accuracy.

[0091] Graph G9a shows the results for the first condition. Graph G9b shows the results for the second condition. Graph G9c shows the results for the third condition. Graph G9d shows the results for the comparison condition. The reference condition is a prediction (comparative example) obtained using a learning model with a loss function that does not include a gradient term, as used in Example 1.

[0092] Comparing graphs G9a, G9b, and G9c with graph G9d, it was found that, similar to the results of Example 1, the predictions made by the learning model M obtained using a loss function L that includes a gradient term were more accurate than those made by the learning model obtained using a loss function that does not include a gradient term. Furthermore, comparing graphs G9a, G9b, and G9c, it was observed that the prediction accuracy tended to be higher under conditions where the term evaluating absolute error was given more weight than the term including gradient (Condition 2: G9b, Condition 3: G9c) compared to the condition where the term evaluating absolute error and the term including gradient were given equal weight (Condition 1: G9a). In other words, it was suggested that a larger weight for the term evaluating absolute error than for the term including gradient may yield better prediction values. [Explanation of symbols]

[0093] 1. Physical field prediction device 2. Learning device (Physical field prediction model learning device) 3 Reasoning device 4. Numerical analysis device 21 Input section 22 Prediction Section 23 Evaluation Department 24 Adjustment part 25 Output section

Claims

1. A physical field prediction model learning device for predicting a physical field composed of physical quantities defined for each of the multiple nodes in a region represented by multiple nodes, An input unit that accepts training data including training data representing the initial state of the physical field and ground truth data obtained as a result of applying the training data to the governing equations that govern the physical field, A prediction unit that obtains predicted values ​​of physical quantities defined for each of the plurality of nodes by applying the training data to a learning model for predicting the physical field, An evaluation unit that evaluates the predicted value of the physical quantity using the aforementioned correct data, The system includes an adjustment unit that adjusts the parameters of the learning model based on the results of the evaluation unit, The evaluation unit evaluates the predicted value of the physical quantity using a loss function, The loss function includes the first term, the second term, and the third term, The first term includes the difference between the true value of the physical quantity given as the correct answer data and the predicted value of the physical quantity output by the prediction unit. The second term includes, as a term relating to the gradient, a value obtained by substituting the predicted value of the physical quantity into the governing equation which includes an element for differentiating the physical quantity, The third term refers to a physical field prediction model learning device that includes values ​​obtained by substituting predicted values ​​of the physical quantities into boundary conditions given as Dirichlet boundary conditions.

2. The loss function includes the first term, The physical field prediction model learning device according to claim 1, wherein the first item includes a true value relating to the gradient of the physical quantity and a predicted value relating to the gradient of the physical quantity.

3. The physical field prediction model learning apparatus according to claim 1 or 2, wherein the loss function includes a weighted average calculation of the first term, the second term, and the third term.

4. A method for learning a physical field prediction model for predicting a physical field composed of physical quantities defined for each of the multiple nodes in a region represented by multiple nodes, Computers A step of receiving training data that includes training data representing the initial state of the physical field and ground truth data obtained as a result of applying the training data to the governing equations that govern the physical field, The steps include: applying the training data to a learning model for predicting the physical field to obtain predicted values ​​of physical quantities defined for each of the multiple nodes; A step of evaluating the predicted value of the physical quantity using the aforementioned correct data, The steps include adjusting the parameters of the learning model based on the results of the evaluation step, The evaluation step described above involves evaluating the predicted value of the physical quantity using a loss function, The loss function includes the first term, the second term, and the third term, The first term includes the difference between the true value of the physical quantity given as the correct data and the predicted value of the physical quantity obtained as a result of the step of obtaining the predicted value, The second term includes, as a term relating to the gradient, a value obtained by substituting the predicted value of the physical quantity into the governing equation which includes an element for differentiating the physical quantity, The third paragraph relates to a physical field prediction model learning method, which includes values ​​obtained by substituting predicted values ​​of the physical quantities into boundary conditions given as Dirichlet boundary conditions.

5. A physical field prediction model learning program for predicting a physical field composed of physical quantities defined for each of the multiple nodes in a region represented by multiple nodes, A step of receiving training data that includes training data representing the initial state of the physical field and ground truth data obtained as a result of applying the training data to the governing equations that govern the physical field, The steps include: applying the training data to a learning model for predicting the physical field to obtain predicted values ​​of physical quantities defined for each of the multiple nodes; A step of evaluating the predicted value of the physical quantity using the aforementioned correct data, The computer is made to perform the following steps: adjusting the parameters of the learning model based on the results of the evaluation step; The evaluation step described above involves evaluating the predicted value of the physical quantity using a loss function, The loss function includes the first term, the second term, and the third term, The first term includes the difference between the true value of the physical quantity given as the correct data and the predicted value of the physical quantity obtained as a result of the step of obtaining the predicted value, The second term includes, as a term relating to the gradient, a value obtained by substituting the predicted value of the physical quantity into the governing equation which includes an element for differentiating the physical quantity, The third term refers to a physical field prediction model learning program that includes values ​​obtained by substituting predicted values ​​of the physical quantities into boundary conditions given as Dirichlet boundary conditions.

Images