Data-driven multi-physics time-domain non-intrusive model reduction method and system

Abstract

The application discloses a data-driven multi-physical field time domain non-intrusive model reduction method and system, and the method is as follows: S1, obtaining a multi-physical field thermal simulation high-dimensional data set through high-precision finite elements; S2, starting from the multi-physical field thermal simulation high-dimensional data set, converting the data set into an array format, and adopting a data dimension reduction method to compress and reduce the spatial dimension; S3, determining boundary conditions and heat source terms in the multi-physical field thermal simulation calculation process as external conditions of the data-driven process; S4, performing operator reasoning on the data set of S2, the boundary conditions and the heat source terms of S3, obtaining a reduction equation operator, and constructing a new reduction equation model from the reduction operator; and S5, solving the new low-dimensional reduction equation through Euler iteration, reconstructing a high-dimensional temperature field distribution, and verifying the accuracy and effectiveness of the new reduction model. The application combines control equations and physical information with scientific machine learning algorithms, has physical interpretability, and improves the time domain calculation efficiency of the multi-physical field.

Description

Technical Field

[0001] This invention belongs to the field of multiphysics thermal simulation technology, and specifically relates to a data-driven, non-intrusive multiphysics temporal model reduction method and system in a three-dimensional integrated microsystem and a 2.5D chip packaging structure based on Chiplet, which realizes data dimensionality reduction and accelerates computation. Background Technology

[0002] In the field of multiphysics coupled simulation, large-scale linear equation systems with hundreds of thousands to tens of millions of degrees of freedom are ultimately required to solve. This involves enormous computational demands and memory consumption, making conventional solution methods inefficient. Model Reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. Its purpose is to reduce the computational complexity of such problems. By reducing the relevant state space dimension or degrees of freedom of the model, an approximation of the original model is calculated, often referred to as a reduced-order model (ROM).

[0003] Scientific machine learning is a technical field that applies machine learning to the modeling and analysis of physical systems. Traditional physical system modeling relies on physical equations and prior knowledge, which typically requires significant time and effort. Scientific machine learning, by using machine learning methods, can learn and simulate the dynamic behavior and properties of physical systems from experimental data without requiring explicit equations. Scientific machine learning can aid in the development of reduced-order models by learning low-dimensional structures and properties from high-dimensional data, thereby improving the accuracy and reliability of reduced-order models.

[0004] Data-driven order reduction modeling methods use input-output data from the original high-fidelity model to construct a reduced-order model that accurately represents the underlying system. Non-intrusive order reduction modeling methods use data to learn low-dimensional operators; currently, common approaches involve learning the mapping of partial differential equation state coefficients from inputs to low-dimensional output data using surrogate models or similar methods. This mapping can be parameterized using Kriging models, radial basis functions, neural networks, or nearest neighbor regression. However, these works do not incorporate knowledge of the system's governing equations into the parameterization learning process, resulting in computational inefficiency. Summary of the Invention

[0005] To address the aforementioned problems, this invention provides a data-driven, non-intrusive, time-domain model order reduction method and system for multiphysics. It employs data processing methods to compress the spatial dimension of the state solution dataset generated by the training model. By using the compressed dataset, it learns low-dimensional order reduction operators, thus learning the mapping of partial differential equation state coefficients from input to low-dimensional output data. This allows for the learning and simulation of the dynamic behavior and characteristics of physical systems from experimental data without requiring access to high-dimensional operators of explicit equations.

[0006] The present invention adopts the following technical solution:

[0007] The steps of the data-driven, multiphysics, time-domain non-invasive model reduction method include:

[0008] S1. Obtaining high-dimensional datasets: Obtaining high-dimensional datasets for multiphysics thermal simulations using high-precision finite element methods;

[0009] S2. Data Dimensionality Reduction: Starting from the high-dimensional dataset of multiphysics thermal simulation, the dataset is converted into an array format, and spatial dimension compression and dimensionality reduction are performed using data dimensionality reduction methods such as intrinsic orthogonal decomposition (POD) and singular value decomposition (SVD).

[0010] S3. Determine the boundary conditions and heat source terms in the multiphysics thermal simulation calculation process as external conditions in the data-driven method process;

[0011] S4. Using the operator reasoning method, the dataset obtained in step S2, the boundary conditions and heat source terms obtained in step S3 are subjected to operator reasoning to obtain the reduced-order equation operator, and a new reduced-order equation model is constructed from the reduced-order operator.

[0012] S5. Solve the new low-dimensional reduced-order equations through Euler iteration, reconstruct the high-dimensional temperature field distribution, and verify the accuracy and effectiveness of the new reduced-order model.

[0013] Preferably, in step 2, the specific method for data dimensionality reduction is as follows:

[0014] S2.1. Orthogonal Eigenvalue Decomposition (POD) is a mathematical method for extracting basic information features from discrete data. Its core algorithm is Singular Value Decomposition (SVD). POD is used to compress discrete data. First, the dataset Q... m×n Perform singular value decomposition (SVD), decomposing it into the product of three matrices U, ∑, and V:

[0015]

[0016] Among them, Q m×n For a high-dimensional dataset to be reduced in order, U m×m For a left singular matrix, ∑ m×n It is a singular value matrix, with diagonal elements representing the dataset Q. m×n There are n singular values.

[0017] S2.2.∑ m×n The diagonal of σ represents the singular values. As the number of singular values ​​increases, the corresponding values ​​of the singular values ​​become smaller. Often, the energy contained in the first 10% or even the first 1% of singular values ​​can account for 99% of the total energy of all singular values. The first r singular values ​​are selected according to requirements to ensure:

[0018]

[0019] Where, σ i Represents dataset Q m×n The i-th singular value, where ε represents the cumulative energy loss of the first r singular values, usually ε < 0.1, and usually r << n;

[0020] S2.3. Select the first r singular values ​​for truncation, and use the first r columns of the left singular matrix as the new truncation basis vector matrix. Based on the mathematical meaning of the matrix inner product, project the original high-dimensional dataset into a new low-dimensional space, thus completing the spatial dimensionality compression of the dataset.

[0021]

[0022] Among them, U m×r This indicates that the left singular matrix, truncated based on the first r singular values, is used as the truncated basis vector matrix; using... This represents a low-dimensional dataset under the new truncated basis vectors, completing the compression of the dataset in terms of spatial dimensions.

[0023] Preferably, in step 4, the operator reasoning method process is as follows:

[0024] S4.1. For the problem of solving multiphysics partial differential equations, the governing equations are written in a semi-discrete form after spatial discretization:

[0025]

[0026] q(0)=q0

[0027] t∈[t0,t end (4)

[0028] Where q represents the current state variable, u represents the excitation term, c represents the constant term, and A, H, C, and B represent full-order operators. Let q represent the rate of change of variable q with respect to time t, q(0) represent the state variable of the system at time zero, q0 represent the initial value, t represent time, t0 represent the initial time, and t end Represents the end time;

[0029] S4.2. The invasive results of the original governing equations are generated using the Galerkin projection method:

[0030]

[0031] in, Represents an intrusive order reduction operator.

[0032] R represents the real number field;

[0033] S4.3. This invention employs a data-driven, non-intrusive model reduction method. It learns low-dimensional reduction operators using a compressed dataset, learning the mapping from the input to the state coefficients of the partial differential equations without accessing high-dimensional operators. The dataset matrix and operator matrix are constructed, and operator inference is performed.

[0034]

[0035]

[0036] in, For low-dimensional datasets, is the Kronecker product operator, where D represents the dataset matrix and O represents the operator matrix;

[0037] S4.4. The corresponding order reduction operator is obtained through operator reasoning, and the reduced order equation is:

[0038]

[0039]

[0040] t∈[t0,t end (8)

[0041] in, The goal of learning, in order to obtain the reduced-order data, is to find the most suitable reduction operator. and To minimize the objective function;

[0042] S4.5. The objective function is as follows:

[0043]

[0044] in, Reduced-order data representing the time derivative;

[0045] This invention also discloses a data-driven, multiphysics, time-domain non-invasive model reduction system, which, based on the above method, includes the following modules:

[0046] High-dimensional dataset acquisition module: Acquires high-dimensional datasets for multiphysics thermal simulations using high-precision finite element methods;

[0047] Data dimensionality reduction module: Starting from the high-dimensional dataset of multiphysics thermal simulation, the dataset is converted into an array format, and the spatial dimension is compressed and reduced using data dimensionality reduction methods;

[0048] Adjustment and determination module: Determines the boundary conditions and heat source terms in the multiphysics thermal simulation calculation process, as the external conditions in the data-driven process;

[0049] Operator reasoning module: Performs operator reasoning on the obtained dataset, boundary conditions and heat source terms to obtain reduced-order equation operators, and constructs a new reduced-order equation model from the reduced-order operators;

[0050] Verification module: By solving the new low-dimensional reduced-order equation through Euler iteration, the high-dimensional temperature field distribution is reconstructed, verifying the accuracy and effectiveness of the new reduced-order model.

[0051] Compared with existing model reduction methods, this invention has significant advantages:

[0052] (1) A non-intrusive model reduction method is adopted to learn and simulate the dynamic behavior and characteristics of physical systems from experimental data without accessing the high-dimensional operators of explicit equations, thus possessing physical interpretability.

[0053] (2) Using dimensionality reduction methods such as POD and SVD, the original high-dimensional dataset is reduced to tens of thousands of dimensions, achieving a reduction of more than two orders of magnitude.

[0054] (3) Data-driven approach using physics-based scientific machine learning methods. Due to the introduction of system control equations, the reduced-order model learned by this method has a good generalization effect. Attached Figure Description

[0055] Figure 1 This is a schematic diagram showing the relationship between the number of singular values ​​and the logarithmic function value of singular values ​​when SVD is applied to the temperature field data of the Chiplet 2.5D package structure.

[0056] Figure 2 This is a schematic diagram showing the relationship between the number of singular values ​​and the cumulative energy of singular values ​​when SVD is applied to the temperature field data of the Chiplet 2.5D package structure.

[0057] Figure 3 This is a flowchart illustrating the specific process of the non-invasive model reduction method according to a preferred embodiment of the present invention.

[0058] Figure 4 This is a schematic diagram of the geometric model of a 2.5D chip package structure based on Chiplet;

[0059] Figure 5 This is a comparison of the temperature field distribution of the full-order model and the reduced-order model of the Chiplet 2.5D package structure at t=700s. (a) is the temperature field distribution of the full-order model (unit degC), (b) is the temperature field distribution of the reduced-order model (unit degC), and (c) is the error distribution (unit degC).

[0060] Figure 6This is a comparison of the temperature field distribution of the full-order model and the reduced-order model of the Chiplet 2.5D package structure at t=900s. (a) is the temperature field distribution of the full-order model (unit degC), (b) is the temperature field distribution of the reduced-order model (unit degC), and (c) is the error distribution (unit degC).

[0061] Figure 7 This is a block diagram of a data-driven, multi-physics, time-domain non-intrusive model reduction system according to a preferred embodiment of the present invention. Detailed Implementation

[0062] The following specific implementation examples, in conjunction with the accompanying drawings, illustrate the embodiments of the present invention. Those skilled in the art can intuitively understand other advantages of the present invention from the description herein. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that, unless otherwise specified, the following embodiments and features described therein can be combined with each other.

[0063] like Figure 1-6 The present embodiment provides a data-driven, non-intrusive, time-domain model order reduction method for multiphysics, which includes the following steps:

[0064] S1. Obtaining a High-Dimensional Dataset: A high-dimensional dataset for multiphysics thermal simulation is obtained using high-precision finite element methods. This process involves modeling and solving complex physical fields using the finite element method. First, an accurate model needs to be established, considering material properties, boundary conditions, and initial conditions. Then, the model is discretized into a finite element mesh, where each element represents a small region. Next, the discretized equations are solved to obtain the temperature at each point. These numerical solutions correspond to each data point in the high-dimensional dataset. By meshing and solving over the entire computational domain, a large number of data points can be obtained, forming a high-dimensional dataset.

[0065] S2. Data Dimensionality Reduction: Starting with high-dimensional datasets from multiphysics thermal simulations, the datasets are transformed into array format. Dimensionality reduction methods such as Orthogonal Eigenvalue Decomposition (POD) and Singular Value Decomposition (SVD) are used to compress the spatial dimension. The basic idea of ​​the POD method is to construct an autocorrelation matrix using numerical simulation or experimental data of complex systems, then solve for the eigenvectors of the autocorrelation matrix to obtain the POD reduced-order modes or reduced-order basis functions. Finally, the high-dimensional system is projected onto the subspace spanned by the reduced-order modes corresponding to the first few largest eigenvalues, thus achieving the goal of order reduction.

[0066] For singular values, they are arranged in descending order in the singular value matrix, and the singular values decrease particularly rapidly. In many cases, the sum of the first 10% or even 1% of singular values accounts for more than 99% of the sum of all singular values.

[0067] Step S2 is specifically as follows:

[0068] S2.1. Proper Orthogonal Decomposition (POD) is a mathematical method for extracting basic information features from discrete data. The core of its algorithm is Singular Value Decomposition (SVD). Data compression is performed on discrete data by using Proper Orthogonal Decomposition (POD), firstly through the data set Q m×n performing Singular Value Decomposition (SVD), which can be decomposed into the product of three matrices U, Σ, and V:

[0069]

[0070] Wherein, Q m×n is a high-dimensional dataset to be reduced, U m×m is the left singular matrix, Σ m×n is the singular value matrix, the diagonal elements are the singular values of the dataset Q m×n , with a total of n items;

[0071] S2.2. The diagonal of Σ m×n is the singular value σ. For the relationship between the number of singular values and the logarithmic function value of singular values, refer to Figure 1 , select the first r singular values according to requirements to ensure:

[0072]

[0073] Wherein, σ i represents the i-th singular value of the dataset Q m×n , ε represents the cumulative energy loss of the first r singular values, usually ε < 0.1, and usually r << n. For the relationship between the number of singular values and the cumulative energy value of singular values, refer to Figure 2 ;

[0074] S2.3. Select appropriate first r singular values for truncation, use the first r columns of the left singular matrix as the new truncated basis vector matrix, project the original high-dimensional dataset into the new low-dimensional space according to the mathematical meaning of matrix inner product, and complete the compression of the dataset in spatial dimension.

[0075]

[0076] Wherein, U m×r represents the left singular matrix truncated according to the first r singular values used as the truncated basis vector matrix, and multiplication with the original high-dimensional dataset Q m×n means projecting it onto the space with In a new low-dimensional space with row vectors as basis vectors;

[0077] S3. Determine the boundary conditions and heat source terms in the multiphysics thermal simulation calculation process as external conditions in the data-driven method. Boundary conditions typically include temperature boundary conditions and heat flux boundary conditions. Temperature boundary conditions mainly include fixed temperature values, temperature gradients, or temperature conditions related to other physical phenomena. Heat flux boundary conditions can be fixed heat flux values ​​or heat flux conditions related to heat convection with other objects. Heat source terms typically include internal heat sources and external heat sources. Internal heat sources refer to heating elements or energy-generating regions within the material; their location, size, and heat source intensity need to be clearly defined. External heat sources refer to the heat load provided by the external environment or other objects interacting with the model.

[0078] S4. Using the operator inference method, the reduced-order equation operator is learned from the dataset obtained in step S2, the boundary conditions obtained in step S3, and the heat source term. A new reduced-order equation model is then constructed from the reduced-order operator; specifically as follows:

[0079] S4.1. For typical multiphysics partial differential equation solving problems, the system's governing equations are written in semi-discrete form after spatial discretization:

[0080]

[0081] q(0)=q0

[0082] t∈[t0,t end (4)

[0083] Where q represents the current state variable of the system, u represents the excitation term of the system, c is the constant term, and A, H, C, and B represent the full-order operators of the system;

[0084] S4.2. Traditional Galerkin projection-based methods generate intrusive results for the original system's control equations:

[0085]

[0086] in, The intrusive order reduction operator representing the system,

[0087] Traditional Galerkin projection-based order reduction methods target the original system's control equations themselves, which may require accessing and manipulating the internal data structures and variables of the full-order model, or even modifying the backend source code.

[0088] S4.3. This invention employs a data-driven, non-intrusive model reduction method. It learns low-dimensional reduction operators using a compressed dataset, learning the mapping from the input to the state coefficients of the partial differential equations without accessing high-dimensional operators. The dataset matrix and operator matrix are constructed, and operator inference is performed.

[0089]

[0090]

[0091] in, For low-dimensional datasets, This is the Kronecker product operator;

[0092] S4.4. The corresponding order reduction operator is obtained through operator reasoning, and the reduced order equation is:

[0093]

[0094]

[0095] t∈[t0,t end (8)

[0096] in, For reduced-order data obtained through POD and SVD methods, the learning objective is to find the most suitable reduction operator. and To minimize the objective function;

[0097] S4.5. The objective function is in the following form:

[0098]

[0099] S5. Solve the new low-dimensional reduced-order equations using Euler iterations to reconstruct the high-dimensional temperature field distribution and verify the accuracy and effectiveness of the new reduced-order model; details are as follows:

[0100] S5.1. First, assign the low-dimensional initial system state variable s_initial_hat to the first column of the low-dimensional system state variable s_hat to be solved, as the initial value for Euler iteration;

[0101] S5.2. Calculate the corresponding difference matrix using the difference matrix, the low-dimensional initial system state variable s_initial_hat, and the low-dimensional operator. and The excitation value u is used to iteratively solve for s_hat;

[0102] S5.3. By calculating U rs_hat, reconstructing the high-dimensional temperature field distribution to verify the accuracy and effectiveness of the new reduced-order model; where U r This is to truncate the basis vector matrix.

[0103] like Figure 7 As shown, this embodiment discloses a data-driven, multiphysics-based, time-domain non-intrusive model reduction system. Based on the above method embodiment, it includes the following modules:

[0104] High-dimensional dataset acquisition module: Acquires high-dimensional datasets for multiphysics thermal simulations using high-precision finite element methods;

[0105] Data dimensionality reduction module: Starting from the high-dimensional dataset of multiphysics thermal simulation, the dataset is converted into an array format, and the spatial dimension is compressed and reduced using data dimensionality reduction methods;

[0106] Adjustment and determination module: Determines the boundary conditions and heat source terms in the multiphysics thermal simulation calculation process, as the external conditions in the data-driven process;

[0107] Operator reasoning module: Performs operator reasoning on the obtained dataset, boundary conditions and heat source terms to obtain reduced-order equation operators, and constructs a new reduced-order equation model from the reduced-order operators;

[0108] Verification module: By solving the new low-dimensional reduced-order equation through Euler iteration, the high-dimensional temperature field distribution is reconstructed, verifying the accuracy and effectiveness of the new reduced-order model.

[0109] Other aspects of this embodiment can be found in the above method embodiments.

[0110] In summary, this invention discloses a data-driven, non-intrusive time-domain model reduction method and system for multiphysics. The method includes the following steps: S1, obtaining a multiphysics simulation dataset using a high-precision finite element method; S2, compressing and reducing the spatial dimension of the dataset using intrinsic orthogonal decomposition (POD) and singular value decomposition (SVD) methods; S3, determining the boundary conditions and heat source terms in the multiphysics thermal simulation calculation process as external conditions in the data-driven method process; S4, performing operator inference for the reduction equations using a data-driven scientific machine learning algorithm; S5, reconstructing the time-domain physical field distribution using an Euler iterative solution algorithm. Compared to existing conventional model reduction methods, this invention combines system control equations and physical information with scientific machine learning algorithms, possessing physical interpretability and significantly improving the time-domain computational efficiency of multiphysics.

[0111] Note that the above description is merely a preferred embodiment of the present invention and the technical principles employed. Those skilled in the art will understand that the present invention is not limited to the specific embodiments described herein, and various obvious changes, readjustments, and substitutions can be made without departing from the scope of protection of the present invention. Therefore, although the present invention has been described in detail through the above embodiments, the present invention is not limited to the above embodiments, and may include many other equivalent embodiments without departing from the concept of the present invention, the scope of which is determined by the scope of the appended claims.

Claims

1. A data-driven, non-intrusive time-domain model order reduction method for multiphysics, characterized in that, Includes the following steps: S1. Obtain high-dimensional datasets for multi-physics thermal simulation using high-precision finite element method; S2. Starting from the high-dimensional dataset of multiphysics thermal simulation, the dataset is converted into an array format, and the spatial dimension is compressed and reduced using data dimensionality reduction methods. S3. Determine the boundary conditions and heat source terms in the multiphysics thermal simulation calculation process as external conditions in the data-driven process; S4. Perform operator reasoning on the dataset obtained in step S2, the boundary conditions and heat source terms obtained in step S3 to obtain the reduced-order equation operator, and construct a new reduced-order equation model from the reduced-order operator. S5. Solve the new low-dimensional reduced-order equations through Euler iteration, reconstruct the high-dimensional temperature field distribution, and verify the accuracy and effectiveness of the new reduced-order model; Step S2 specifically includes: S2.

1. By analyzing the dataset Perform singular value decomposition (SVD) to decompose into Product of three matrices: in, For a high-dimensional dataset to be reduced in order, It is a left singular matrix. It is a singular value matrix, with the diagonal elements representing the dataset. The singular values, total item; S2.

2. The diagonal is a singular value. Select according to your needs A singular value to ensure: in, Represents the dataset The A singular value, Indicates the preceding The cumulative energy loss of each singular value, taken as... , ; S2.

3. Before selection The singular values ​​are truncated, and the first singular value of the left singular matrix is ​​used to truncate the data. The columns are used as new truncation basis vector matrices to project the original high-dimensional dataset into a new low-dimensional space, thus compressing the dataset in terms of spatial dimension. in, Indicates based on the previous The left singular matrix truncated by singular values ​​is used as the truncated basis vector matrix, along with the original high-dimensional dataset. Multiplication means projecting it onto a matrix. In a new low-dimensional space with row vectors as basis vectors; Step S5 is as follows: S5.

1. Convert the low-dimensional initial state variables Assigned to the low-dimensional state variables being solved The first column is used as the initial value for the Euler iteration solution; S5.

2. Calculate the corresponding difference matrix, using the difference matrix and low-dimensional initial state variables. Low-dimensional operators and incentive value Perform iterative solution ; S5.

3. Through calculation The high-dimensional temperature field distribution was reconstructed to verify the accuracy and effectiveness of the new reduced-order model; among which... This is to truncate the basis vector matrix.

2. The data-driven, multiphysics-based, time-domain non-invasive model order reduction method according to claim 1, characterized in that, Step S4 specifically includes: S4.

1. For the problem of solving multiphysics partial differential equations, the governing equations are written in a semi-discrete form after spatial discretization: in, Represents the current state variable. c represents the excitation term, and c represents the constant term. Represents a full-order operator. Representative variable Relative to time rate of change, Represents the state variables of the system at time zero. Represents the initial value. Represents time, Represents the initial moment. Represents the end time; S4.

2. The invasive results of the original governing equations are generated using the Galerkin projection method: in, Represents an intrusive order reduction operator. , , , , , Represents the real number field; S4.

3. Construct the dataset matrix and operator matrix, and perform operator inference: in, ∈ For low-dimensional datasets, For Kronek's product operator, Represents a dataset matrix. Represents the operator matrix; S4.

4. The corresponding order reduction operator is obtained through operator reasoning, and the reduced-order equation is: in, The goal of learning, in order to obtain the reduced-order data, is to find the most suitable reduction operator. This achieves the minimum of the objective function; S4.

5. The objective function is as follows: in, Reduced-order data representing the time derivative.

3. A data-driven, multiphysics, time-domain non-invasive model reduction system, based on the method described in any one of claims 1-2, characterized in that... Includes the following modules: High-dimensional dataset acquisition module: Acquires high-dimensional datasets for multiphysics thermal simulations using high-precision finite element methods; Data dimensionality reduction module: Starting from the high-dimensional dataset of multiphysics thermal simulation, the dataset is converted into an array format, and the spatial dimension is compressed and reduced using data dimensionality reduction methods; Adjustment and determination module: Determines the boundary conditions and heat source terms in the multiphysics thermal simulation calculation process, as the external conditions in the data-driven process; Operator reasoning module: Performs operator reasoning on the obtained dataset, boundary conditions and heat source terms to obtain reduced-order equation operators, and constructs a new reduced-order equation model from the reduced-order operators; Verification module: By solving the new low-dimensional reduced-order equation through Euler iteration, the high-dimensional temperature field distribution is reconstructed, verifying the accuracy and effectiveness of the new reduced-order model.

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