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Solving High-dimensional Parametric Elliptic Equation Using Tensor Neural Network

Hongtao Chen, Rui Fu, Yifan Wang, Hehu Xie

TL;DR

This work tackles solving high-dimensional parametric elliptic PDEs with random coefficients by combining a KL expansion with a tensor neural network (TNN) ansatz. The TNN expresses the solution as a sum of rank-one terms across spatial and stochastic dimensions, enabling a quadrature scheme that reduces high-dimensional integrations to products of 1D integrals, and achieving polynomial-in-dimension complexity. A Ritz-type loss (weak or strong form) is minimized via gradient-based optimization to obtain accurate approximations, with boundary conditions incorporated into the architecture. Numerical experiments up to $M=100$ demonstrate accurate, scalable performance across various coefficient decays, validating the method’s potential for uncertainty quantification in high-dimensional SPDEs.

Abstract

In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we can transform the high-dimensional integrations of tensor neural network functions to one-dimensional integrations which can be computed with the classical quadrature schemes with high accuracy. The complexity of its calculation can be reduced from the exponential scale to a polynomial scale. The corresponding machine learning method is designed for solving high-dimensional parametric elliptic equations. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms.

Solving High-dimensional Parametric Elliptic Equation Using Tensor Neural Network

TL;DR

This work tackles solving high-dimensional parametric elliptic PDEs with random coefficients by combining a KL expansion with a tensor neural network (TNN) ansatz. The TNN expresses the solution as a sum of rank-one terms across spatial and stochastic dimensions, enabling a quadrature scheme that reduces high-dimensional integrations to products of 1D integrals, and achieving polynomial-in-dimension complexity. A Ritz-type loss (weak or strong form) is minimized via gradient-based optimization to obtain accurate approximations, with boundary conditions incorporated into the architecture. Numerical experiments up to demonstrate accurate, scalable performance across various coefficient decays, validating the method’s potential for uncertainty quantification in high-dimensional SPDEs.

Abstract

In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we can transform the high-dimensional integrations of tensor neural network functions to one-dimensional integrations which can be computed with the classical quadrature schemes with high accuracy. The complexity of its calculation can be reduced from the exponential scale to a polynomial scale. The corresponding machine learning method is designed for solving high-dimensional parametric elliptic equations. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms.
Paper Structure (11 sections, 4 theorems, 58 equations, 4 figures, 3 tables)

This paper contains 11 sections, 4 theorems, 58 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model problem
  3. Tensor neural network and its quadrature scheme
  4. Tensor neural network architecture
  5. Quadrature scheme for TNN
  6. TNN based machine learning method
  7. Numerical examples
  8. Zero-order coefficients
  9. Variable stochastic coefficients
  10. Exponential decay coefficients
  11. Conclusion

Key Result

Proposition 2.1

(see TodorSchwab). If the exponential decay condition (Exponential_decay) holds, $u=u_M(y, \cdot)$ is the exact solution of (ref:strong) and $M$ is large enough, then the following estimate for all $y \in \Gamma$ and $\alpha \in \mathbb{N}_0^M$ where $\mathbb N_0$ denotes the set of all non-negative integers.

Figures (4)

  • Figure 1: Architecture of TNN. Black arrows mean linear transformation (or affine transformation). Each ending node of blue arrows is obtained by taking the scalar multiplication of all starting nodes of blue arrows that end in this ending node. The final output of TNN is derived from the summation of all starting nodes of red arrows.
  • Figure 2: The relative errors during training process in Example 1.
  • Figure 3: The relative errors during training process in Example 2.
  • Figure 4: The relative errors during training process in Example 3.

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Proposition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.1
  • Remark 3.2