Gaussian sum method for solving multi-body Schrodinger equation through tensor neural network

Abstract

The invention discloses a Gaussian sum method for solving a multi-body Schrodinger equation through a tensor neural network, and relates to the field of quantum physics. According to the method, the kernel decomposition technology of the coulomb kernel is introduced into the framework of solving the multi-body Schrodinger equation through the tensor neural network, and the problem of low calculation efficiency caused by difficult coulomb kernel tensor representation is successfully solved. On the basis of an existing tensor neural network framework, a Gaussian sum decomposition method is introduced for development, a set of complete self-adaptive secondary processing strategy is developed for high-dimensional integral calculation of electron-atomic nucleus and electron-electron interaction with the maximum time and memory overhead, the memory overhead is remarkably reduced, and meanwhile the calculation efficiency is effectively improved.

Description

Technical Field

[0001] This invention relates to the field of quantum physics, and more particularly to a method for solving the Gaussian sum of the multibody Schrödinger equation using tensor neural networks. Background Technology

[0002] Existing algorithms for multibody Schrödinger equations mainly follow three methodological paths:

[0003] The first type of method directly constructs multi-electron wavefunctions, starting with the mean-field approximation of the Hartree-Fock method, and constructs multiple excitation configurations and exponential correlation operators, such as the configuration interaction method (CI) and the coupled cluster method (CC). It can obtain high-precision benchmark calculations, but it faces the computational complexity of higher-order polynomials or even exponentials, making it difficult to extend to complex systems.

[0004] The second type of method starts from electron density and constructs an approximation of the Kohn–Sham equation, known as density functional theory (DFT). However, its approximation faces a "Jacob ladder"-like computational bottleneck, namely, the construction of the exchange-correlation functional is subject to a balance between physical constraints, mathematical rigor, and computational cost, making it difficult to perform high-precision calculations.

[0005] The third type of method is based on statistical simulation and machine learning:

[0006] 1. The quantum Monte Carlo method calculates high-dimensional integrals through sampling, which can flexibly handle strongly correlated systems. However, it still faces problems such as statistical fluctuations and slow convergence speed. In addition, it is sensitive to the singularity of the Coulomb nucleus and often requires careful design of pseudopotentials to replace it.

[0007] 2. Machine learning methods, leveraging the powerful expressive ability of neural networks to solve high-dimensional problems, utilize neural networks to perform ab initio calculations of the multi-body Schrödinger equation. Specifically, this can be further divided into the following two approaches:

[0008] 2.1 Designing network structures based on physical priors. For example, FermiNet remains the most accurate machine learning algorithm to date. However, its variational Monte Carlo integral scheme is still accompanied by statistical fluctuations, and its convergence speed is slow. It often relies on large-scale parallel computing, lengthy training and pre-training processes, and large memory requirements, resulting in huge computational resource consumption.

[0009] 2.2 Designing network structures based on function approximation theory. For example, tensor neural networks (TNNs) are designed based on the Sobolev space H... m (Ω) Tensor Isomorphism Starting from this point, construct the tensor product type basis functions and the approximation subspace:

[0010]

[0011] in For the domain of tensor product type, i = 1, ..., dN represents dN sub-neural networks, θ i Let represent the network parameters of the i-th sub-neural network, where parameter p represents the dimension of the approximating subspace, also known as the rank parameter of the TNN. Then, using the Galerkin method, the optimal solution is found in the subspace. A schematic diagram of the network structure is shown below. Figure 1 As shown. TNN can handle high-dimensional integrals by combining its unique tensor structure with Gaussian quadrature scheme, avoiding statistical fluctuations and achieving fast convergence, thus overcoming the curse of dimensionality. However, it is difficult to obtain a tensor-type variable-separated structure for the Coulomb kernel in the Schrödinger equation. The traditional spherical harmonic function expansion method used is still based on the approximation method obtained by orthogonal polynomials. This not only fails to effectively handle kernel singularity but also makes the method computationally inefficient and memory-intensive.

[0012] Therefore, those skilled in the art are dedicated to developing a Gaussian sum method for solving the multi-body Schrödinger equation using tensor neural networks. This invention introduces the Coulomb kernel decomposition technique into the framework of solving the multi-body Schrödinger equation using tensor neural networks, and proposes a complete adaptive quadratic processing strategy based on the Gaussian sum method for high-dimensional integral calculations of electron-nucleus and electron-electron interactions, which incur the greatest time and memory overhead. This strategy significantly reduces memory consumption while effectively improving computational efficiency. Summary of the Invention

[0013] In view of the above-mentioned deficiencies of the prior art, the technical problem to be solved by the present invention is the computational inefficiency caused by the difficulty in representing the Coulomb kernel tensor in the Schrödinger equation.

[0014] To achieve the above objectives, this invention provides a method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks, comprising the following steps:

[0015] Step 1: Initialize the tensor neural network;

[0016] Step 2: Construct the approximation subspace;

[0017] Step 3: Coulomb kernel Gaussian function and approximation. Based on the bandwidth, the Gaussian function is divided into three parts: long, medium and short ranges. The mass matrix and stiffness matrix are calculated.

[0018] Step 4: Solve the linear generalized eigenvalue problem to obtain the minimum feature pairs;

[0019] Step 5: Assemble the loss function and update the neural network parameters;

[0020] Step 6: Repeat steps 2 through 5 until the maximum number of iterations is reached to obtain an approximate solution.

[0021] Further, in step 1, the maximum number of iterations and the learning rate are input, and the factor before the self-antisymmetric penalty term is written into the loss function.

[0022] Furthermore, in step 3, the stiffness matrix is ​​decomposed into the sum of the kinetic energy component, the potential energy component between atomic nuclei, the potential energy component between electrons and atomic nuclei, and the potential energy component between electrons.

[0023] Furthermore, the potential energy portion between the electron and the atomic nucleus is decomposed into Coulomb nuclei using the Gaussian sum method.

[0024] Furthermore, the potential energy between electrons is divided into long-range interaction, short-range interaction, and medium-range interaction, with the division criteria explicitly determined by the target accuracy and complexity analysis.

[0025] Furthermore, the long-range interaction portion is decomposed using Chebyshev polynomials for quadratic tensor decomposition.

[0026] Furthermore, the short-range interaction component is approximately a Dirac delta function.

[0027] Furthermore, in the mid-range interaction part, the interaction kernel matrix is ​​modeled by reducing its order using eigenvalue decomposition.

[0028] Furthermore, in step 5, the neural network parameters are updated using stochastic gradient descent.

[0029] Furthermore, in step 6, the minimum feature pair and tensor neural network model are output, and the approximate solution of the wave function is obtained by linearly combining the feature vectors and basis functions.

[0030] The difficulty in representing the Coulomb kernel tensor leads to high computational complexity, low efficiency, and large memory requirements in existing tensor neural network solutions to the multi-body Schrödinger equation. This invention addresses this issue by introducing Gaussian sum decomposition, a kernel decomposition technique. Specifically, it first presents a Gaussian function approximation of the Coulomb kernel. The separability of the Gaussian function across its dimensions transforms the high-dimensional interactions still present in the tensor neural network framework into a product of one-dimensional interactions. Furthermore, based on different bandwidths, the Gaussian function is divided into long, medium, and short-range components, and corresponding quadratic processing methods are designed. This invention's Gaussian sum method can efficiently transform the Coulomb kernel into a summation of Gaussian functions according to the target accuracy. The separability of the Gaussian function across its dimensions enables dimensionality reduction of the interaction integral. The bandwidth determines the range and properties of the interaction forces expressed by the Gaussian function; therefore, a corresponding interaction force decomposition strategy can be designed based on the bandwidth size. This invention can reduce the computational complexity by an order of magnitude, decrease memory usage by an order of magnitude, and more than double the computational efficiency.

[0031] The existing technique of using tensor neural networks to solve the multi-body Schrödinger equation, which employs spherical harmonic function expansion of the Coulomb kernel, still falls under the category of traditional orthogonal polynomial expansions and struggles to handle kernel singularities. The Gaussian sum method of this invention provides a uniformly smooth expression for the Coulomb kernel, transforming the singular Coulomb kernel into a smooth Gaussian function summation, effectively handling the singularity. This Gaussian sum method transforms the singularity into a Gaussian function with a small bandwidth, allowing it to be approximated as a Dirac delta function using quadratic processing, thus achieving a solution. The convergence and error estimation of this invention are reliably guaranteed.

[0032] Existing techniques for solving the many-body Schrödinger equation using tensor neural networks have limitations in extending to computations of complex atomic and macromolecular systems. This invention introduces a Gaussian sum decomposition tensor neural network, whose parameters can be explicitly given based on the target accuracy, resulting in a significant improvement in simulation efficiency for complex systems. The Gaussian sum method of this invention, as an efficient kernel function approximation technique, allows for explicit control of the upper bound of its uniform approximation error. This invention can be extended to efficient single-card operation for the first two atomic periods.

[0033] Compared with the prior art, the present invention has the following obvious substantive features and significant advantages:

[0034] 1. This invention introduces a Gaussian sum method to address the computational inefficiency of tensor neural networks caused by the difficulty of representing Coulomb kernels in tensors. Combined with a secondary processing strategy designed based on the bandwidth properties of different Gaussian functions, it can be naturally compatible with multi-GPU parallel technology, which is beneficial for large-scale system computation.

[0035] 2. This invention introduces the Gaussian sum method, a kernel decomposition technique, to provide an efficient low-rank expression for the Coulomb kernel, effectively reducing computational complexity and simultaneously reducing memory requirements by an order of magnitude, thereby more than doubling computational efficiency.

[0036] 3. This invention can be applied to solving the multibody Schrödinger equation using tensor neural networks, a problem that is difficult to handle efficiently with complex atomic or molecular systems under current technology. This invention has broad application potential and value in major cutting-edge scientific and technological fields such as catalyst guidance, optical material screening, materials design, and optical simulation.

[0037] The following will further explain the concept, specific structure, and technical effects of the present invention in conjunction with the accompanying drawings, so as to fully understand the purpose, features, and effects of the present invention. Attached Figure Description

[0038] Figure 1 This is a schematic diagram of a tensor neural network structure;

[0039] Figure 2 This is a flowchart of the multi-body Schrödinger equation algorithm based on the Gaussian sum method and tensor neural network of this invention;

[0040] Figure 3 This is the ground-state energy error diagram of the helium atom system calculation example test of this invention;

[0041] Figure 4 This is the radial probability density distribution of the wave function obtained from the numerical test of the helium atom system of this invention;

[0042] Figure 5 This is the ground state energy error diagram of the lithium atom system calculation example test of this invention;

[0043] Figure 6 This is the radial probability density distribution of the wave function obtained from the lithium atom system numerical test of this invention;

[0044] Figure 7 This is the ground state energy error diagram of the beryllium atom system calculation example of this invention;

[0045] Figure 8 This is the radial probability density distribution of the wave function obtained from the beryllium atom system numerical test of this invention. Detailed Implementation

[0046] The following description, with reference to the accompanying drawings, illustrates several preferred embodiments of the present invention to make its technical content clearer and easier to understand. The present invention can be embodied in many different forms, and the scope of protection of the present invention is not limited to the embodiments mentioned herein.

[0047] In the accompanying drawings, components with the same structure are indicated by the same numerical designation, and components with similar structures or functions are indicated by similar numerical designations. The dimensions and thicknesses of each component shown in the drawings are arbitrary, and the present invention does not limit the dimensions and thicknesses of each component. To make the illustrations clearer, the thickness of some components has been appropriately exaggerated in the drawings.

[0048] This invention addresses the inefficiency of existing tensor neural network methods for solving the Coulomb kernel in the multi-body Schrödinger equation by proposing a solution based on the Gaussian sum method. Specifically, firstly, a Gaussian function approximation of the Coulomb kernel is explicitly given. Leveraging the separability of the Gaussian function across dimensions and the consistency of the Gaussian sum approximation accuracy, the high-dimensional interaction integral is transformed into a one-dimensional interaction integral, effectively handling the impact of kernel singularity on computation. Secondly, based on the bandwidth properties of each Gaussian function, the invention divides each term into long-range, medium-range, and short-range interaction components, designing a corresponding quadratic decomposition scheme for further low-rank decomposition. This significantly reduces computational overhead and memory requirements, enabling efficient operation in a single-GPU environment. This solution balances accuracy and efficiency and demonstrates greater application potential in complex systems and excited-state energy calculations. Figure 2 A detailed flowchart of the present invention is presented.

[0049] Steps S1-S2: First, input the maximum number of iterations L and the learning rate γ, and simultaneously add the pre-term factor for the self-opposite symmetry penalty term. The loss function is written as a global parameter. Next, using a tensor neural network composed of dN fully connected sub-neural networks from existing technologies, initialization is performed first, and then an approximation subspace as shown in equation (1) from existing technologies is constructed. This transforms the numerical solution of the high-dimensional partial differential equation into the following optimization problem:

[0050]

[0051] Where <·|·> denotes the inner product in high-dimensional space. The parameters of a tensor neural network are denoted as [1, N1, N2, ..., N]. L [,p] represents the number of neurons in the input, hidden, and output layers of each subnetwork, respectively. In this expression, the Hamiltonian operator... The energy operator is determined by the many-body Schrödinger equation itself:

[0052]

[0053] Where Δ i For the Laplace operator with respect to the coordinates of the i-th particle, {r i =(x i ,y i ,z i )} and {R k =(X k ,Y k Z k {Q} represents the three-dimensional coordinates of the electrons and the atomic nucleus, respectively. k} represents the charge number of the corresponding atomic nucleus, N ↑ and N ↓ σ represents the number of electrons in the up and down spin directions, respectively. ↑ and σ ↓ The mapping of particle indices corresponding to spin directions is represented by the commutation operator T. ij Ψ(r1,…,r i ,…,r j ,…,r N )=Ψ(r1,…,r j ,…,r i ,…,r N ) should meet

[0054]

[0055] This can be achieved by adjusting the penalty factor. To impose self-opasymmetry restrictions.

[0056] Step S3: Element-by-element calculation of the mass matrix <Ψ|Ψ> can be performed quickly using a tensor neural network; for the stiffness matrix... Element-by-element calculations show that the Hamiltonian operator, as an energy operator, allows the stiffness matrix to be naturally decomposed into the sum of the kinetic energy component, the potential energy component between atomic nuclei, the potential energy component between electrons and atomic nuclei, and the potential energy component between electrons:

[0057] 1. Kinetic Energy Component: Tensor Neural Networks Guarantee Gradient Operators Even after the effect, the mass matrix can still be compared, enabling efficient and fast calculations.

[0058]

[0059] 2. Atomic Nucleus and its component: Under the Born-Oppenheimer approximation, the atomic nuclei of a many-body system are in fixed positions. Therefore, for free electrons, the mass matrix can be analogously used to achieve efficient and fast calculations.

[0060]

[0061] 3. Electrons and Atomic Nucleus: Since the position of the atomic nucleus is fixed, the Coulomb nucleus is decomposed using the Gaussian sum method:

[0062]

[0063] Where b represents the scaling factor, and σ controls the bandwidth of each Gaussian function. This indicates that the mean is 0 and the variance is b. 2l σ 2 The Gaussian distribution of this term. This high-dimensional integral also has a tensor product representation of the one-dimensional integral, and therefore can be computed efficiently and quickly by analogy with the mass matrix.

[0064]

[0065] 4. Electron-electron interaction: After decomposing the Coulomb nucleus using the Gaussian sum method, this high-dimensional integral still contains electron-electron interaction terms:

[0066]

[0067] The tensor product form of a one-dimensional integral cannot be directly obtained. Note that each Gaussian function has different low-rank properties based on its bandwidth; therefore, this invention designs an adaptive partitioning based on different bandwidths, and uses different quadratic decomposition strategies in each partition. Specifically, this invention divides the Coulomb potential energy between electrons into three interaction parts—long-range, medium-range, and short-range—and processes them sequentially. The parameter partitioning criteria are explicitly determined by the analysis of target accuracy and complexity.

[0068] 4.1 Long-range interaction: Gaussian functions with large bandwidth exhibit very smooth changes, and orthogonal polynomial expansion is computationally efficient. Therefore, this invention employs Chebyshev polynomials for quadratic tensor decomposition:

[0069]

[0070] in Denotes the set of all natural numbers, with coefficients... T can be obtained through pre-calculation. m (·) represents an m-th order Chebyshev polynomial, which ultimately yields a one-dimensional tensor product expression of the Coulomb kernel, thus allowing for efficient and rapid computation by analogy with the mass matrix.

[0071] 4.2 Short-range interaction: As the bandwidth approaches zero, the Gaussian function converges to the Dirac delta function according to the distribution:

[0072]

[0073] The Dirac delta function can transform convolution into function values, thereby achieving dimensionality reduction. Therefore, this invention can reduce the dimensionality of the interaction between two electrons using the Dirac delta function approximation, based on the target accuracy.

[0074]

[0075] This ultimately yields a one-dimensional tensor product representation of the Coulomb kernel. This can then be used as an analogy to the mass matrix, enabling efficient and rapid computation.

[0076] 4.3 Mid-range Interaction Part: Although this part only has a small number of Gaussian functions, the efficiency of using orthogonal polynomial expansion for the corresponding Gaussian function is poor, and approximating it with the Dirac function has the limitation of excessive error, making it the most difficult part to handle. This invention starts from its symmetry and uses eigenvalue decomposition technology to reduce the order of the interaction kernel matrix model:

[0077]

[0078] in The interaction kernel matrix is ​​represented by matrix V, which represents the orthogonal matrix in the eigenvalue decomposition. Λ is the eigenvalue diagonal matrix. The eigenvalue matrix can be truncated according to the target accuracy, thereby reducing the order of the model and achieving efficient and fast calculation of this part.

[0079] Steps S4 - S5: After obtaining the stiffness matrix and mass matrix at the current iteration step, use the Galerkin method to transform it into a linear generalized eigenvalue problem in an algebraic system. The minimum eigenvalue can be used as an approximation of the ground state energy to assemble the loss function as shown in equation (2) in Steps S1 - S2. The minimum eigenvector is used as an approximate solution of the wave function. This process can be efficiently solved using the built - in functions in the torch library. Finally, use the Adam optimizer in the torch library to update the neural network parameters in a stochastic gradient descent manner.

[0080] Step S6: Loop the operations from Step S2 to Step S5 until the maximum number of iteration steps is reached. Finally, output the minimum eigenpair and the tensor neural network model, and linearly combine the eigenvector with the basis function to obtain an approximate solution of the wave function.

[0081] Complexity analysis: Let K represent the number of nodes of one - dimensional Gaussian integration, N represent the number of electrons, M represent the number of atomic nuclei, d represent the physical space dimension (which is 3), and p represent the rank parameter of the tensor neural network. The following computational complexity estimates for each part can be obtained:

[0082] 1. Computational complexity of the integral of the wave function inner product term:

[0083] 2. Computational complexity of the integral of the kinetic energy term:

[0084] 3. Potential energy term between atomic nuclei and atomic nuclei: This term is the wave function inner product matrix in 1 multiplied by a pre - computable constant.

[0085] 4. Potential energy term between atomic nuclei and electrons: where T ae represents the number of Gaussian sum truncations of the Coulomb kernel between the nucleus and the electron.

[0086] 5. Potential energy term between electrons and electrons:

[0087] Long - range interaction part: where represents the number of truncations in the long - range part of the Gaussian sum of the Coulomb kernel between the nucleus and the electron, and R represents the truncation number of the Chebyshev expansion of the Gaussian function.

[0088] Short - range interaction part:

[0089] Mid - range interaction part: where represents the number of truncations in the mid - range part of the Gaussian sum of the Coulomb kernel between the nucleus and the electron, and K EVD <K represents the number of nodes of one - dimensional Gaussian integration after model rank reduction using the eigenvalue decomposition technique.

[0090] Traditional methods for high-precision calculations suffer from the curse of dimensionality, resulting in a computational complexity of O(n). The computational complexity of directly solving the multi-body Schrödinger equation using tensor neural networks is O(n). Where T L K is the cutoff number of the spherical harmonic function expansion, and K also represents the number of surrogate points in each dimension of the corresponding method. In comparison, this invention better handles the singularity of Coulomb interaction while reducing the prefactor of computational complexity by more than an order of magnitude. For a comparison of accuracy, efficiency, and memory overhead, please refer to the embodiments and Table 1.

[0091] This invention innovatively introduces a Coulomb kernel decomposition technique into the framework of tensor neural networks for solving the multi-body Schrödinger equation, successfully addressing the computational inefficiency caused by the difficulty in representing the Coulomb kernel tensor. Specifically, this invention develops a Gaussian sum decomposition method based on the existing tensor neural network framework. Steps S1-2 and S4-6 use the original TNN framework, while step S3 is the high-dimensional integral calculation kernel. This invention proposes a complete adaptive quadratic processing strategy based on the Gaussian sum method for the high-dimensional integral calculation of electron-nucleus and electron-electron interactions (corresponding to parts 3 and 4 of step S3), which has the highest time and memory overhead. This strategy significantly reduces memory overhead while effectively improving computational efficiency.

[0092] Example:

[0093] Helium atom testing system: As a two-electron system, helium atoms do not require self-antisymmetry. A high-dimensional cube with a side length of 20a.u. can be used for region truncation, where au represents the Bohr radius. Numerical modeling is performed using a tensor neural network of size [1, 50, 50, 60] and 340 Gaussian-Legendal integration points in one dimension. The Gaussian sum decomposition parameters are set to b = 1.3, σ = 1, l ∈ [-150, 80]. The final result is a piecewise optimized error map (e.g., ...). Figure 3 (as shown) and the radial probability density distribution of the wave function (as shown) Figure 4 (As shown). The test was conducted on a single NVIDIA A800 card.

[0094] Lithium and beryllium atom testing systems: As multi-electron systems, lithium and beryllium atoms require a self-antisymmetric penalty term to satisfy the Pauli exclusion principle. Both systems were tested on a single NVIDIA A800 GPU. For lithium atoms, a high-dimensional cube with a side length of 24 a.u. was used for region truncation. A tensor neural network of size [1, 50, 50, 60] with 340 Gaussian-Legendary integration points was used for numerical modeling. Training was completed with a self-antisymmetric penalty term with a factor of 50. The Gaussian sum decomposition parameters were set to b = 1.3, σ = 1, l ∈ [-150, 80]. The final piecewise optimization error map (e.g.) was obtained. Figure 5 (as shown) and the radial probability density distribution of the wave function (as shown) Figure 6 (As shown); the precision requirements for beryllium atoms are relatively low, and the Gaussian and decomposition parameters are taken as b = 1.4, σ = 1, l ∈ [-130, 80]. A high-dimensional cube with a side length of 30 a.u. is used for region truncation, and a tensor neural network with a scale of [1, 50, 50, 60] and 310 single-dimensional Gaussian-Legendary integration points are used for numerical modeling. At the same time, a self-antisymmetric penalty term with a front factor of 50 is added to complete the training, and finally the piecewise optimization error map is obtained (as shown). Figure 7 (as shown) and the radial probability density distribution of the wave function (as shown) Figure 8 (As shown).

[0095] The memory overhead of the above tests is extremely small; a single card (NVIDIA A800 with 80GB of VRAM) can easily handle the computation and has the potential to support larger-scale system computations. A table comparing the accuracy and efficiency of each system is shown below. Compared to existing algorithms that use tensor neural networks based on spherical harmonic function expansion to solve the multi-body Schrödinger equation, this invention can improve the accuracy of the beryllium atom ground state energy by four orders of magnitude. Simultaneously, while improving the accuracy of other systems, it reduces memory usage by one order of magnitude and more than doubles computational efficiency.

[0096] Table 1: Accuracy and efficiency of the multi-body Schrödinger equation tensor neural network algorithm based on the Gaussian sum method of this invention

[0097]

[0098] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A method for solving the Gaussian sum of the multibody Schrödinger equation using tensor neural networks, characterized in that, Includes the following steps: Step 1: Initialize the tensor neural network; Step 2: Construct the approximation subspace; Step 3: Coulomb kernel Gaussian function and approximation. Based on the bandwidth, the Gaussian function is divided into three parts: long, medium and short ranges. The mass matrix and stiffness matrix are calculated. Step 4: Solve the linear generalized eigenvalue problem to obtain the minimum feature pairs; Step 5: Assemble the loss function and update the neural network parameters; Step 6: Repeat steps 2 through 5 until the maximum number of iterations is reached to obtain an approximate solution.

2. The method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks as described in claim 1, characterized in that, In step 1, the maximum number of iterations and the learning rate are input, and the factor before the self-antisymmetric penalty term is written into the loss function.

3. The method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks as described in claim 1, characterized in that, In step 3, the stiffness matrix is ​​decomposed into the sum of the kinetic energy component, the potential energy component between atomic nuclei, the potential energy component between electrons and atomic nuclei, and the potential energy component between electrons.

4. The method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks as described in claim 3, characterized in that, The potential energy between the electron and the atomic nucleus is decomposed into Coulomb nuclei using the Gaussian sum method.

5. The method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks as described in claim 3, characterized in that, The potential energy between electrons is divided into long-range interaction, short-range interaction, and medium-range interaction, and the division criteria are explicitly determined by the analysis of target accuracy and complexity.

6. The method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks as described in claim 5, characterized in that, The long-range interaction part is decomposed into quadratic tensors using Chebyshev polynomials.

7. The method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks as described in claim 5, characterized in that, The short-range interaction component is approximately a Dirac delta function.

8. The method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks as described in claim 5, characterized in that, In the mid-range interaction part, the interaction kernel matrix is ​​reduced in order using eigenvalue decomposition.

9. The method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks as described in claim 1, characterized in that, In step 5, the neural network parameters are updated using stochastic gradient descent.

10. The method for solving the Gaussian sum of the multi-body Schrödinger equation using tensor neural networks as described in claim 1, characterized in that, In step 6, the minimum feature pair and tensor neural network model are output, and the approximate solution of the wave function is obtained by linearly combining the feature vectors and basis functions.

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