Interpolating Neural Network-Tensor Decomposition (INN-TD): a scalable and interpretable approach for large-scale physics-based problems

TL;DR

This work presents INN-TD, a scalable and interpretable framework that combines Convolutional-Hierarchical Deep Neural Network (C-HiDeNN) local interpolation with tensor decomposition to model high-dimensional PDEs. It enables data-driven training, data-free solving, and inverse optimization within a single architecture, delivering superior accuracy and efficiency compared to traditional neural surrogates and tensor-based baselines. The method leverages FE-inspired locality, automatic Dirichlet handling, and a mode-wise tensor structure to circumvent the curse of dimensionality while preserving interpretability through explicit mesh and mode semantics. Findings indicate strong performance across parametric PDEs, high-dimensional Poisson and SP-T PDEs, and inverse problems, suggesting practical impact for industrial-scale physics-based simulations demanding precision and speed.

Abstract

Deep learning has been extensively employed as a powerful function approximator for modeling physics-based problems described by partial differential equations (PDEs). Despite their popularity, standard deep learning models often demand prohibitively large computational resources and yield limited accuracy when scaling to large-scale, high-dimensional physical problems. Their black-box nature further hinders the application in industrial problems where interpretability and high precision are critical. To overcome these challenges, this paper introduces Interpolating Neural Network-Tensor Decomposition (INN-TD), a scalable and interpretable framework that has the merits of both machine learning and finite element methods for modeling large-scale physical systems. By integrating locally supported interpolation functions from finite element into the network architecture, INN-TD achieves a sparse learning structure with enhanced accuracy, faster training/solving speed, and reduced memory footprint. This makes it particularly effective for tackling large-scale high-dimensional parametric PDEs in training, solving, and inverse optimization tasks in physical problems where high precision is required.

Figures (17)

• Figure 1: C-HiDeNN approaximation based on locally supported basis functions
• Figure 2: Interpretable design of INN-TD: (a) the 1D meshes used in INN-TD are only refined at the location where nonlinearity happens with larger $s$ and $p$ used to achieve higher-order smoothness (b) point-wise absolute error between INN-TD and the exact solution
• Figure 3: Problem statement
• Figure 4: Analytical solution for 2D cases.
• Figure 5: Solving the 2D Poisson's Eq. using INN-TD with different number of model parameters and hyperparameters $s$ and $p$. $a$=20 is used for all cases. The statistics are obtained by repeating each case for 10 times and the shaded area represents 1 standard deviation (a) convergence of INN-TD using different $s$ and $p$ (b) statistics of total computational time (c) statistics of GPU VRAM usage