Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces

TL;DR

This paper introduces a spectral polynomial surrogate model for Sobolev spaces that efficiently solves a wide range of PDEs, outperforming PINNs in accuracy and speed, and enabling solutions without high-performance computing.

Contribution

The paper develops a spectral polynomial surrogate model (PSM) for Sobolev spaces, providing a convex variational formulation for PDEs that is computationally efficient and more accurate than existing neural network approaches.

Findings

PSMs achieve exponential convergence rates.

PSMs outperform PINNs in accuracy and runtime.

PSMs enable solving PDEs without high-performance computing.

Abstract

We introduce a novel spectral, finite-dimensional approximation of general Sobolev spaces in terms of Chebyshev polynomials. Based on this polynomial surrogate model (PSM), we realise a variational formulation, solving a vast class of linear and non-linear partial differential equations (PDEs). The PSMs are as flexible as the physics-informed neural nets (PINNs) and provide an alternative for addressing inverse PDE problems, such as PDE-parameter inference. In contrast to PINNs, the PSMs result in a convex optimisation problem for a vast class of PDEs, including all linear ones, in which case the PSM-approximate is efficiently computable due to the exponential convergence rate of the underlying variational gradient descent. As a practical consequence prominent PDE problems were resolved by the PSMs without High Performance Computing (HPC) on a local machine. This gain in efficiency is complemented by an increase of approximation power, outperforming PINN alternatives in both accuracy and runtime. Beyond the empirical evidence we give here, the translation of classic PDE theory in terms of the Sobolev space approximates suggests the PSMs to be universally applicable to well-posed, regular forward and inverse PDE problems.