Spectral-Informed Neural Networks Outperform Spectral Methods in High-dimensional PDEs

2026-07-15

AI Summary

This work addresses the longstanding challenge of solving high-dimensional partial differential equations (PDEs), which has been hindered by the curse of dimensionality: conventional spectral methods suffer from rapidly escalating computational costs, while physics-informed neural networks (PINNs) often lack sufficient accuracy and efficiency. The paper proposes a modified Spectral-Informed Neural Network (Modified SINN) that directly approximates unknown spectral coefficients in the spectral domain by introducing a harmonic-analysis-driven coefficient decay scaling and a basis function embedding mechanism. This approach eliminates the need for spatial derivative computations and substantially reduces memory consumption. The method outperforms sparse-grid spectral methods in moderate dimensions and significantly surpasses PINNs in both steady-state and time-dependent high-dimensional PDE problems, achieving markedly higher accuracy and computational efficiency.

Abstract

For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems ($4 \leq d \lesssim 10$), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-dimensional problems ($d\gg 10$), spectral methods suffer frome the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising approach to overcome this challenge, offering scalability to high dimensions, but often suffer from limited accuracy and efficiency. Recently proposed spectral-informed neural networks (SINNs) combine spectral methods with PINNs, operating directly in the spectral domain to avoid spatial derivative computations and to reduce memory consumption. In this work, we introduce Modified SINNs, which integrate coefficient decay scaling and basis embeddings motivated by harmonic analysis to enhance accuracy in high-dimensional problems and enable accurate approximation of unknown spectral coefficients. Numerical experiments on steady and time-dependent partial differential equations demonstrate that Modified SINNs outperform sparse grid spectral methods on middle-dimensional problems with incomplete spectral information and achieve superior accuracy compared to PINNs on high-dimensional problems.

Problem

• high-dimensional PDEs
• curse of dimensionality
• spectral methods
• physics-informed neural networks
• spectral accuracy

Innovation

• spectral-informed neural networks
• high-dimensional PDEs
• coefficient decay scaling
• basis embeddings
• physics-informed neural networks