Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities
Samuel Lanthaler, Roberto Molinaro, Patrik Hadorn, Siddhartha Mishra
Abstract
Discontinuous solutions arise in a large class of hyperbolic and advection-dominated PDEs. This paper investigates, both theoretically and empirically, the operator learning of PDEs with discontinuous solutions. We rigorously prove, in terms of lower approximation bounds, that methods which entail a linear reconstruction step (e.g. DeepONets or PCA-Nets) fail to efficiently approximate the solution operator of such PDEs. In contrast, we show that certain methods employing a non-linear reconstruction mechanism can overcome these fundamental lower bounds and approximate the underlying operator efficiently. The latter class includes Fourier Neural Operators and a novel extension of DeepONets termed shift-DeepONets. Our theoretical findings are confirmed by empirical results for advection equations, inviscid Burgers’ equation and the compressible Euler equations of gas dynamics.
BibTeX
@inproceedings{
lanthaler2023nonlinear,
title={Nonlinear Reconstruction for Operator Learning of {PDE}s with Discontinuities},
author={Samuel Lanthaler and Roberto Molinaro and Patrik Hadorn and Siddhartha Mishra},
booktitle={The Eleventh International Conference on Learning Representations },
year={2023},
url={https://openreview.net/forum?id=CrfhZAsJDsZ}
}