NONLINEAR RECONSTRUCTION FOR OPERATOR LEARNING OF PDES WITH DISCONTINUITIES

Abstract

A large class of hyperbolic and advection-dominated PDEs can have solutions with discontinuities. 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. DeepONet or PCA-Net) fail to efficiently approximate the solution operator of such PDEs. In contrast, we show that certain methods employing a nonlinear 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 DeepONet termed shift-DeepONet. Our theoretical findings are confirmed by empirical results for advection equation, inviscid Burgers' equation and compressible Euler equations of aerodynamics.

1. INTRODUCTION

Many interesting phenomena in physics and engineering are described by partial differential equations (PDEs) with discontinuous solutions. The most common types of such PDEs are nonlinear hyperbolic systems of conservation laws (Dafermos, 2005) , such as the Euler equations of aerodynamics, the shallow-water equations of oceanography and MHD equations of plasma physics. It is well-known that solutions of these PDEs develop finite-time discontinuities such as shock waves, even when the initial and boundary data are smooth. Other examples include the propagation of waves with jumps in linear transport and wave equations, crack and fracture propagation in materials (Sun & Jin, 2012), moving interfaces in multiphase flows (Drew & Passman, 1998) and motion of very sharp gradients as propagating fronts and traveling wave solutions for reaction-diffusion equations (Smoller, 2012) . Approximating such (propagating) discontinuities in PDEs is considered to be extremely challenging for traditional numerical methods (Hesthaven, 2018) as resolving them could require very small grid sizes. Although bespoke numerical methods such as high-resolution finitevolume methods, discontinuous Galerkin finite-element and spectral viscosity methods (Hesthaven, 2018) have successfully been used in this context, their very high computational cost prohibits their extensive use, particularly for many-query problems such as UQ, optimal control and (Bayesian) inverse problems (Lye et al., 2020) , necessitating the design of fast machine learning-based surrogates. As the task at hand in this context is to learn the underlying solution operator that maps input functions (initial and boundary data) to output functions (solution at a given time), recently developed operator learning methods can be employed in this infinite-dimensional setting (Higgins, 2021). These methods include operator networks (Chen & Chen, 1995) and their deep version, DeepONet (Lu et al., 2019; 2021) , where two sets of neural networks (branch and trunk nets) are combined in a

