Neural Spectral Methods: Self-supervised learning in the spectral domain
This work addresses the problem of efficiently solving PDEs for computational science and engineering, offering a novel method that is not incremental but introduces a new training strategy.
The paper tackles solving parametric Partial Differential Equations (PDEs) by introducing Neural Spectral Methods, which uses a spectral loss based on Parseval's identity to improve training efficiency and inference speed, resulting in performance gains of one to two orders of magnitude over previous machine learning approaches and a 10× speed increase compared to numerical solvers at the same accuracy.
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.