In-Person Oral presentation / top 25% paper

Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities

Samuel Lanthaler · Roberto Molinaro · Patrik Hadorn · Siddhartha Mishra

[ Abstract ] [ Livestream: Visit Oral 6 Track 1: Theory ]
Wed 3 May 7:10 a.m. — 7:20 a.m. PDT

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.

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