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Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities

by S. Lanthaler and R. Molinaro and P. Hadorn and S. Mishra

(Report number 2022-42)

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.

Keywords: Operator Learning; Discontinuous solutions of PDEs

@Techreport{LMHM22_1030,
  author = {S. Lanthaler and R. Molinaro and P. Hadorn and S. Mishra},
  title = {Nonlinear Reconstruction for Operator Learning of
PDEs with Discontinuities},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-42},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-42.pdf },
  year = {2022}
}

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