A data-driven multi-physics time-domain model reduction method for electromagnetic systems

Abstract

The present invention discloses a method for reducing the time-domain model of multiple physical fields in a data-driven electromagnetic system. The innovation of this method lies in combining scientific machine learning with model reduction for the first time and using it to solve electromagnetic system problems. First, the spatial dimension of the data set is reduced by the orthogonal projection (POD) method, which can achieve a data reduction of more than three orders of magnitude; then, the amount of data is reduced in the time dimension by an adaptive frame selection method; then, a data-driven method of scientific machine learning is used to learn a new model from the reduced data set. The reduced-order model can replace different applications of the full-order model to achieve more than a thousand times of computational acceleration. The method of the present invention has the characteristics of high scalability and small error, is particularly suitable for small sample data-driven situations, and can greatly improve the time-domain computing efficiency of multiple physical fields in electromagnetic systems.

Description

Technical Field

[0001] The present invention relates to a data-driven multi-physics field time-domain model reduction method in electromagnetic systems such as high-power semiconductor devices, integrated microsystems and integrated circuits, and is particularly aimed at the electric (magnetic)-thermal-mechanical multi-physics field coupling scenario in electromagnetic systems to achieve accelerated calculation. Background Art

[0002] The coupled solution of multi-physics fields in electromagnetic systems such as high-power semiconductor devices, integrated microsystems and integrated circuits often faces the problems of huge computational complexity, high memory usage and slow solution convergence. To this end, the present invention proposes a data-driven time domain model reduction method for multi-physics fields in electromagnetic systems, which can realize accelerated calculation of multi-physics fields and significantly reduce memory requirements.

[0003] Prior art, Chinese invention patent "Model reduction method for droop grid-connected inverter network" (publication number: CN113098065A) discloses a model reduction method for droop grid-connected inverter network, which relates to the field of droop grid-connected inverter control technology. This invention is to solve the problem that traditional slow coherence methods are difficult to apply to distributed power generation networks. This invention is aimed at the existing widely used three-phase droop control grid-connected inverter, and proposes a model reduction method for droop grid-connected inverter network. This method can apply the slow coherence algorithm to the droop control inverter network, and can improve the grouping results of the traditional slow coherence algorithm, and further improve the accuracy of the reduced-order model. However, this invention cannot solve the model reduction problem in multi-physics field finite element (FEM) simulation in electromagnetic systems such as high-power semiconductor devices, integrated microsystems and integrated circuits.

[0004] The literature (M.Wang, H.-X.Li, X.Chen, and Y.Chen, “Deep learning-based model reduction for distributed parameter systems,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol.46, no.12, pp.1664–1674, 2016) also uses a non-invasive method that can be effective when the control equations are unknown, but there is a problem of high training cost.

[0005] The method of the present invention compresses the data set in advance, which greatly reduces the training cost; at the same time, the control equation is introduced, but it does not need to access high-dimensional operators, which improves the generalization of the calculation method. Summary of the invention

[0006] In view of the defects in the prior art, the purpose of the present invention is to provide a method for reducing the order of multi-physics field time-domain models in a data-driven electromagnetic system. The method comprises the following steps:

[0007] Step 1: Starting from the existing multi-physics field simulation data set in the electromagnetic system, the data set is converted into an array format, and the spatial dimension of the data set is compressed using the intrinsic orthogonal decomposition method. The compressed data set is used as a new frame added to the data column in the second step;

[0008] Step 2: Adopt an adaptive frame selection method to compress the data column representing the time dimension. The adaptive frame selection method includes: analyzing the sensitive information introduced by each new frame to confirm whether to delete it from the data column. The compressed data column is used as a training set.

[0009] Step 3: Confirm the boundary conditions and source terms in the multi-physics calculation process of the electromagnetic system as external conditions in the data-driven process;

[0010] Step 4: Using a data-driven approach, learn a new reduced-order model from the training set obtained in step 2;

[0011] Step 5: Test the effectiveness of the new reduced-order model. If the error is large, add an appropriate regularization coefficient and retrain.

[0012] Step 6: Repeat steps 4 and 5 until the error between the new reduced-order model and the full-order model reaches the requirement, then stop training.

[0013] Specifically, in the first step, the step of performing the first data compression on the data using the intrinsic orthogonal decomposition method includes: using singular value decomposition SVD or random singular value decomposition RSVD, and compressing the data rows representing the spatial dimensions.

[0014] Specifically, the first step is:

[0015] For dataset A m×n Perform singular value decomposition:

[0016]

[0017] Among them A m×n is the data set to be reduced, Σ m×n The elements on the diagonal are all data sets A m×n The singular values ​​of , there are n singular values ​​in total;

[0018] Select the first k singular values ​​to ensure:

[0019]

[0020] Among them, σ i Represents dataset Am×n The i-th singular value of , ε represents the proportion of the first k singular values ​​to all singular values, and ε>0.9;

[0021] So far, using Represents dataset A m×n , completing the compression of the data set in the spatial dimension.

[0022] Specifically, the adaptive frame selection method in the second step is:

[0023] The error introduced by the n+1th frame is:

[0024]

[0025] In the formula, e n+1 is the error introduced in the n+1th frame, u n is the nth frame data, is the derivative of the nth frame data, Δt is the time interval, u n+1 is the actual n+1th frame data;

[0026] Determine the error between the n+1th frame and the nth frame. If the error introduced by the n+1th frame satisfies the following formula, then keep the frame. Otherwise, remove the frame from the training set:

[0027]

[0028] Among them, k is the upper limit of error control, θ is a hyperparameter, τ represents the order of the equation, ‖e n+1 ‖2 represents the bi-norm of the error introduced in the n+1th frame;

[0029] Add an upper bound on the time interval:

[0030] Δt<=α×Δt min

[0031] Among them, α is the upper boundary coefficient, Δt min is the minimum time interval between frames;

[0032] The method of frame selection is: taking the nth frame as the benchmark, if the n+1th frame is removed from the training set, then calculate the error introduced from the nth frame to the n+2th frame, and so on, until the error introduced by the latest frame reaches the condition to retain the frame, then add the frame to the training set, and start the next frame selection based on this frame; if no frame between the nth frame and the n+αth frame is added to the training set, then add the n+αth frame to the training set, and use the n+αth frame as the benchmark for the next frame selection.

[0033] Specifically, the fourth step is:

[0034] The corresponding reduced-order operator is learned using a data-driven method. The reduced-order equation is:

[0035]

[0036]

[0037] t∈[t0,t f ]

[0038] in For the reduced-order data obtained by the POD method, the learning goal is to find the most suitable reduction operator and Achieve the minimum objective function;

[0039] The objective function is as follows:

[0040]

[0041] in, is the frame data after reduction, s j is time t j External incentives.

[0042] Specifically, the fifth step is:

[0043] Different regularization coefficients λ1 and λ2 are introduced for the first-order term and the second-order term respectively. The objective function after the introduction of the regularization coefficient is defined as:

[0044]

[0045] λ1 is the regularization coefficient introduced for the first-order term, and λ2 is the regularization coefficient introduced for the second-order term;

[0046] The upper bound of the data obtained by introducing ordinary differential solution is O = ω × max|u i，j |; When the maximum value of the solution of the reduced-order ordinary differential equation exceeds O, it means that the regularization coefficient needs to be adjusted. Through multiple adjustments, the maximum value of the solution of the reduced-order ordinary differential equation is guaranteed to be less than O; where ω is a hyperparameter, max|u i，j | is the maximum value of all frames in the original data.

[0047] Compared with the prior art, the present invention has the following significant advantages: (1) It performs large-scale order reduction on data sets in both time and space dimensions, achieving a reduction in data set size of more than three orders of magnitude and a computational acceleration of more than a thousand times. (2) It adopts the scientific machine learning (SciML) method for data-driven learning to obtain new reduced-order models. Due to the introduction of control equations, it can have good generalization properties. (3) It is possible to flexibly select learning variables and only focus on the quantities of interest. (4) It can effectively solve the difficulties of combining multi-physics field calculation problems with machine learning in three-dimensional electromagnetic systems. (5) The calculation method of the present invention has the characteristics of high scalability and small error, and is particularly suitable for small sample data-driven situations, and can greatly improve the time domain calculation efficiency of multi-physics fields in electromagnetic systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 This is a schematic diagram of singular values ​​when POD is applied to IGBT force field data.

[0049] Figure 2 It is a flow chart of a specific method of the present invention.

[0050] Figure 3 It is a flowchart of scientific machine learning.

[0051] Figure 4 This is a schematic diagram of the geometric model of a high-power semiconductor device IGBT chip.

[0052] Figure 5 It is a comparison diagram of the thermal field distribution obtained by the IGBT full-order model and the reduced-order model, where (a) is the thermal field distribution of the full-order model (unit: K), (b) is the thermal field distribution of the reduced-order model (unit: K), and (c) is the error distribution.

[0053] Figure 6 The force field distribution of the full-order model and the reduced-order model of the IGBT is compared, where (a) is the force field distribution of the full-order model (unit N / m 2 ), (b) is the force field distribution of the reduced-order model (unit N / m 2 ), (c) is the error distribution. DETAILED DESCRIPTION

[0054] The present invention is further described in detail below in conjunction with the accompanying drawings. The following examples will help those skilled in the art to further understand the present invention, but are not intended to limit the present invention in any form. It should be noted that, for those of ordinary skill in the art, several changes and improvements can be made without departing from the concept of the present invention. These all belong to the protection scope of the present invention.

[0055] The first step of the present invention is to perform proper orthogonal decomposition (POD) on the data set.

[0056] It is a mathematical method for extracting characteristic information of discrete data. The purpose of the POD method is to describe the multidimensional random process in a low-dimensional approximate manner and to extract the characteristic parameters of the complex random process. The basic idea is to decompose the random quantity into a set of basis functions determined by its own characteristics. The principle of determining the basis function is to make the lowest-order mode term contain the most energy in each decomposition process.

[0057] For the singular values ​​of a data set, they are similar to the eigenvalues ​​in eigendecomposition. They are also arranged from large to small in the singular value matrix, and the decrease of singular values ​​is particularly fast. In many cases, the sum of the first 1% of singular values ​​accounts for more than 99% of the sum of all singular values. In other words, the related matrix system can be approximately described by the largest k singular values ​​and the corresponding left and right singular vectors, that is,

[0058]

[0059] Where k is much smaller than n, a large matrix A can be represented by three small matrices U m×k ,Σ k×k ,V T k×n Jointly expressed.

[0060] Among them A m×n is the original data set, which can be used within the allowable error range. It is expressed as the compressed form; Σ m×n The diagonal of is the obtained singular value σ, and the first k singular values ​​are selected to ensure:

[0061]

[0062] ε is generally greater than 0.99.

[0063] The second step of the present invention is adaptive frame selection. It determines whether to add a frame to the training set based on the error control mechanism. This is an adaptive time step method similar to solving ODE, which makes error judgments on existing data sets. The error calculation is performed on the information of each frame to characterize the amount of information brought by the frame that is different from the previous one. For frames with less new information, they are deleted from the training set.

[0064] For the n+1th frame, the error introduced is:

[0065]

[0066] In the formula, e n+1 is the error introduced in the n+1th frame, u n is the nth frame data, is the derivative introduced by the nth frame data, Δt is the time interval, u n+1is the original data of the n+1th frame;

[0067] Based on the nth frame, the data distribution of the n+1 frame can be predicted. The error introduced between the two is the main basis for determining whether to remove it from the training set:

[0068]

[0069] Where k is the upper boundary of error control. Frames with errors exceeding k are considered meaningful data and retained in the training set. θ is a hyperparameter. Different values ​​are set for different data sets. τ represents the order of the equation. In this case, it is selected as 2. ‖e n+1 ‖2 represents the binary norm of the error introduced in the n+1th frame.

[0070] In addition, we add an upper bound on the time interval, that is,

[0071] Δt<=α×Δt min . (5)

[0072] Among them, α is the upper boundary coefficient, which generally does not exceed 100, Δt min is the minimum time interval between frames. The frame selection method is as follows: taking the nth frame as the reference, when the n+1th frame is removed from the training set, the error introduced from the nth frame to the n+2th frame is calculated, and so on; until the error introduced by the latest frame reaches the condition for retaining the frame (all newly introduced frames are compared with the nth frame, and the formula (4) is used to determine whether to retain it), the frame is added to the training set, and the next frame selection is started with this frame as the reference; if no frame between the nth frame and the n+αth frame is added to the training set, the current frame selection is stopped, the n+αth frame is added to the training set, and the next frame selection is performed with the n+αth frame as the reference.

[0073] The third step of the present invention is to confirm the boundary conditions and source terms in the multi-physics field calculation process of the electromagnetic system as external conditions in the data-driven process. This step is processed in a conventional manner in the art.

[0074] The fourth step of the present invention is data-driven scientific machine learning. For the typical second-order multi-physics field partial differential equation solution problem in electromagnetic systems, the control equation of the system is spatially discretized and written in semi-discrete form as:

[0075]

[0076] Where u(t) represents the current state variable of the system, s(t) is the input of the system, which is generally determined by boundary conditions and excitation terms, n is a constant term, and A, B, C, and D are the full-order operators of the system. represents the Kronecker product, which is a tensor product. t0, t fIt is the system initial time and system final time.

[0077] Using the traditional Galerkin Projection method will produce an intrusive result of the original equation:

[0078]

[0079] where u r (t) represents the reduced-order state variable of the system to be determined, s(t) is the input of the system, which is generally determined by boundary conditions and excitation terms, and n r is a constant term, A r ,B r ,C r ,D r is the operator after the system is reduced in order.

[0080] It can be seen that the object of reduction in traditional order reduction methods is the original control equation itself, which requires access to source code and high-dimensional operators.

[0081] The non-intrusive model reduction method in the present invention does not need to access the full-order operator, and can learn the reduced-order model from the data set. First, it is clear that the reduced-order control equation and the full-order equation have the same polynomial form, so the corresponding reduced-order operator can be learned using a data-driven method. The reduced-order equation is:

[0082]

[0083] in For the reduced-order data obtained by the POD method, the learning goal is to find the most suitable reduction operator and Achieve the minimum objective function.

[0084] The objective function is expressed as:

[0085]

[0086] in, is the j-th frame data after reduction, s j is time t j External incentives.

[0087] In this calculation process, errors may be introduced. In order to avoid overfitting, appropriate regularization terms are added. For this purpose, the objective function is defined as:

[0088]

[0089] By introducing two hyperparameters, namely, Introducing the regularization coefficient λ1, the quadratic term The regularization coefficient λ2 is introduced to minimize the final error. The final optimization result is a reduced-order ordinary differential equation operator. The ultimate goal is to minimize the error between the result of solving the reduced-order ordinary differential equation and the original data. There is a difference between the optimization of the operator and the optimization of the final result of the ordinary differential solution, which is mainly caused by the error. For this reason, it is necessary to introduce a regularization coefficient and introduce the upper boundary O=ω×max|u of the data obtained by the ordinary differential solution. i，j |. When the maximum value of the result obtained from solving the reduced-order ordinary differential equation exceeds O, it means that the regularization coefficient needs to be adjusted. Through multiple adjustments, the maximum value of the solution of the reduced-order ordinary differential equation is guaranteed to be less than O; ω is a hyperparameter, which is generally slightly greater than 1. max|u i，j | is the maximum value of all frames in the original data.

[0090] The traditional full-order model and the reduced-order model of the present invention are used to simulate the thermal field and force field distribution respectively. We apply a fixed external current as a heat source at the position of the IGBT chip and the bonding wire. Due to the thermal effect of the current, the temperature rises. Due to the thermal expansion effect, the stress distribution diagram is obtained. The equation involved is the current continuity equation:

[0091]

[0092] Where J is the applied current density, is the charge change. The current and electric field distribution of the chip can be obtained. Then through the Joule heat equation:

[0093] Q(r,t)=E 丅 J. (12)

[0094] Get the heat generated by the current Q.

[0095] Then, by solving the heat conduction equation, we can get the temperature field distribution:

[0096]

[0097] Where κ(T), ρ, c represent thermal conductivity, density and heat capacity respectively. Q(r,t) is the heat source.

[0098] The elasticity equation is:

[0099]

[0100] The meaning of each character is:

[0101]

[0102] Substituting the thermal strain equation into the elastic mechanics equation, we can get the stress distribution:

[0103] Thermal strain equation:

[0104] ε T =αΔT=α(TT ref ) (15)

[0105] Figure 5 The thermal field distribution comparison diagram of the IGBT full-order model and the reduced-order model, where 5(a) is the thermal field distribution of the full-order model, 5(b) is the thermal field distribution of the reduced-order model, and 5(c) is the error distribution. It can be seen that the error between the full-order result and the reduced-order result of the thermal field is very small, and this method has achieved effective results.

[0106] Figure 6 Comparison of force field distributions obtained by the full-order model and the reduced-order model of IGBT, where 6(a) is the force field distribution of the full-order model, 6(b) is the force field distribution of the reduced-order model, and 6(c) is the error distribution. It can be seen that the error between the full-order result and the reduced-order result of the force field is very small, and this method has achieved effective results.

[0107] Figure 5 and Figure 6 The error calculation method for the results shown in is:

[0108]

[0109] Where 'F' represents full-order model data and 'R' represents reduced-order model data.

[0110] The above describes the specific embodiments of the present invention. It should be understood that the present invention is not limited to the above specific embodiments, and those skilled in the art may make various changes or modifications within the scope of the claims, which will not affect the essence of the present invention.

Claims

1. A method for reducing the order of multi-physics field time-domain models in a data-driven electromagnetic system, characterized in that: The steps include: Step 1: Starting from the existing multi-physics field simulation data set in the electromagnetic system, the data set is converted into an array format, and the spatial dimension of the data set is compressed using the intrinsic orthogonal decomposition method. The compressed data set is used as a new frame added to the data column in the second step; Step 2: Adopt the adaptive frame selection method to compress the data columns representing the time dimension; The adaptive frame selection method is as follows: by analyzing the sensitive information introduced by each new frame, confirm whether to delete it from the data column; the compressed data column is used as the training set; Step 3: Confirm the boundary conditions and source terms in the multi-physics calculation process of the electromagnetic system as external conditions in the data-driven process; Step 4: Using a data-driven approach, learn a new reduced-order model from the training set obtained in step 2; Step 5: Test the effectiveness of the new reduced-order model. If the error is large, add an appropriate regularization coefficient and retrain. Step 6: Repeat steps 4 and 5 until the error between the new reduced-order model and the full-order model reaches the requirement, then stop training; The adaptive frame selection method in the second step is specifically as follows: The error introduced by the n+1th frame is: In the formula, e n+1 is the error introduced in the n+1th frame, u n is the nth frame data, is the derivative of the nth frame data, Δt is the time interval, u n+1 is the actual n+1th frame data; Determine the error between the n+1th frame and the nth frame. If the error introduced by the n+1th frame satisfies the following formula, then keep the frame. Otherwise, remove the frame from the training set: Among them, k is the upper limit of error control, θ is a hyperparameter, τ represents the order of the equation, and ||e n+1 ||2 represents the binary norm of the error introduced in the n+1th frame; Add an upper bound on the time interval: Δt<=α×Δt min Among them, α is the upper boundary coefficient, Δt min is the minimum time interval between frames; The frame selection method is as follows: taking the nth frame as the reference, if the n+1th frame is removed from the training set, then calculate the error introduced from the nth frame to the n+2th frame, and so on, until the error introduced by the latest frame reaches the condition to retain the frame, then add the frame to the training set, and start the next frame selection with this frame as the reference; if no frame between the nth frame and the n+αth frame is added to the training set, then add the n+αth frame to the training set, and use the n+αth frame as the reference for the next frame selection; The fourth step is as follows: The corresponding reduced-order operator is learned using a data-driven method. The reduced-order equation is: in For the reduced-order data obtained by the POD method, the learning goal is to find the most suitable reduction operator and Achieve the minimum objective function; The objective function is as follows: Among them, u j is the frame data after reduction, s j is time t j External incentives.

2. The method for reducing the order of multi-physics field time-domain model in a data-driven electromagnetic system according to claim 1, characterized in that: In the first step, the step of performing the first data compression on the data by using the intrinsic orthogonal decomposition method includes: using singular value decomposition SVD or random singular value decomposition RSVD, and compressing the data rows representing the spatial dimensions.

3. The method for reducing the order of multi-physics field time-domain model in a data-driven electromagnetic system according to claim 1, characterized in that: The first step is: For dataset A m×n Perform singular value decomposition: Among them A m×n is the data set to be reduced, Σ m×n The elements on the diagonal are all data sets A m×n The singular values ​​of , there are n singular values ​​in total; Select the first k singular values ​​to ensure: Among them, σ i Represents dataset A m×n The i-th singular value of , ε represents the proportion of the first k singular values ​​to all singular values, and ε>0.9; So far, using Represents dataset A m×n , completing the compression of the data set in the spatial dimension.

4. The method for reducing the order of multi-physics field time-domain model in a data-driven electromagnetic system according to claim 1, characterized in that: The fifth step is as follows: Different regularization coefficients λ1 and λ2 are introduced for the first-order term and the second-order term respectively. The objective function after the introduction of the regularization coefficient is defined as: λ1 is the regularization coefficient introduced for the first-order term, and λ2 is the regularization coefficient introduced for the second-order term; The upper bound of the data obtained by introducing ordinary differential solution is O = ω × max|u i，j |; When the maximum value of the solution of the reduced-order ordinary differential equation exceeds 0, it means that the regularization coefficient needs to be adjusted. Through multiple adjustments, the maximum value of the solution of the reduced-order ordinary differential equation is ensured to be less than 0; where ω is a hyperparameter, max|u i，j | is the maximum value of all frames in the original data.

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