Neural Spectral Methods: Self-supervised learning in the spectral domain

TL;DR

Neural Spectral Methods introduce a spectral loss-based approach for solving parametric PDEs that outperforms existing machine learning and numerical methods in speed and accuracy, with constant inference cost.

Contribution

The paper proposes a novel spectral loss strategy for neural spectral methods, improving training efficiency and inference speed for PDE solutions.

Findings

Outperforms previous ML approaches by 10-100x in speed and accuracy

Maintains constant inference cost regardless of domain resolution

Achieves 10x faster performance than numerical solvers at same accuracy

Abstract

We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral coefficients. In contrast to current machine learning approaches which enforce PDE constraints by minimizing the numerical quadrature of the residuals in the spatiotemporal domain, we leverage Parseval's identity and introduce a new training strategy through a \textit{spectral loss}. Our spectral loss enables more efficient differentiation through the neural network, and substantially reduces training complexity. At inference time, the computational cost of our method remains constant, regardless of the spatiotemporal resolution of the domain. Our experimental results demonstrate that our method significantly outperforms previous machine learning approaches in terms of speed and accuracy by one to two orders of magnitude on multiple different problems. When compared to numerical solvers of the same accuracy, our method demonstrates a $10\times$ increase in performance speed.