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Convolutional Hierarchical Deep Learning Neural Networks-Tensor Decomposition (C-HiDeNN-TD): a scalable surrogate modeling approach for large-scale physical systems

Jiachen Guo, Chanwook Park, Xiaoyu Xie, Zhongsheng Sang, Gregory J. Wagner, Wing Kam Liu

TL;DR

This work tackles the challenge of scaling surrogate models to extremely large space-time PDEs without expensive offline data generation. It introduces C-HiDeNN-TD, a priori surrogates built from a finite-element–inspired C-HiDeNN basis and a tensor decomposition that expresses multivariate solutions as products of univariate factors, enabling direct computation of surrogate vectors from the PDE. The approach demonstrates convergence on a 2D Poisson problem and showcases large-scale transient diffusion, achieving substantial speedups, dramatically lower memory, and minimal storage compared to traditional methods, with the capability to handle grids up to $51{,}200^4$. This method notes potential extensions to space-time-parameter spaces and nonlinear PDEs like Navier–Stokes, offering a path toward exa-scale surrogates for complex physical systems.

Abstract

A common trend in simulation-driven engineering applications is the ever-increasing size and complexity of the problem, where classical numerical methods typically suffer from significant computational time and huge memory cost. Methods based on artificial intelligence have been extensively investigated to accelerate partial differential equations (PDE) solvers using data-driven surrogates. However, most data-driven surrogates require an extremely large amount of training data. In this paper, we propose the Convolutional Hierarchical Deep Learning Neural Network-Tensor Decomposition (C-HiDeNN-TD) method, which can directly obtain surrogate models by solving large-scale space-time PDE without generating any offline training data. We compare the performance of the proposed method against classical numerical methods for extremely large-scale systems.

Convolutional Hierarchical Deep Learning Neural Networks-Tensor Decomposition (C-HiDeNN-TD): a scalable surrogate modeling approach for large-scale physical systems

TL;DR

This work tackles the challenge of scaling surrogate models to extremely large space-time PDEs without expensive offline data generation. It introduces C-HiDeNN-TD, a priori surrogates built from a finite-element–inspired C-HiDeNN basis and a tensor decomposition that expresses multivariate solutions as products of univariate factors, enabling direct computation of surrogate vectors from the PDE. The approach demonstrates convergence on a 2D Poisson problem and showcases large-scale transient diffusion, achieving substantial speedups, dramatically lower memory, and minimal storage compared to traditional methods, with the capability to handle grids up to . This method notes potential extensions to space-time-parameter spaces and nonlinear PDEs like Navier–Stokes, offering a path toward exa-scale surrogates for complex physical systems.

Abstract

A common trend in simulation-driven engineering applications is the ever-increasing size and complexity of the problem, where classical numerical methods typically suffer from significant computational time and huge memory cost. Methods based on artificial intelligence have been extensively investigated to accelerate partial differential equations (PDE) solvers using data-driven surrogates. However, most data-driven surrogates require an extremely large amount of training data. In this paper, we propose the Convolutional Hierarchical Deep Learning Neural Network-Tensor Decomposition (C-HiDeNN-TD) method, which can directly obtain surrogate models by solving large-scale space-time PDE without generating any offline training data. We compare the performance of the proposed method against classical numerical methods for extremely large-scale systems.
Paper Structure (8 sections, 15 equations, 3 figures)

This paper contains 8 sections, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Method
  3. C-HiDeNN
  4. C-HiDeNN Tensor decomposition
  5. Results
  6. Convergence study: Poisson equation
  7. Large scale transient diffusion analysis
  8. Conclusion and future work

Figures (3)

  • Figure 1: Convergence study of Poisson equation
  • Figure 2: Comparison of C-HiDeNN-TD and finite difference method (FDM)
  • Figure 3: C-HiDeNN structure. (a) 1D C-HiDeNN basis function is constructed by aggregating the information from all the segments patch domain into the current segment (b) Neural network structure of C-HiDeNN basis function: the first 2 hidden layers are mapping the input to piece-wise continuous function $N$, which satisfies the Kronecker delta property; the 3rd layer is the nonlinear part that acts as the convolution patch function $W$, which satisfies the reproducing condition; the output field variable $u^{h,e}$ has higher-order smoothness and automatically satisfy the Dirichlet boundary condition thanks to the Kronecker delta property.