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A large class of hyperbolic and advection-dominated PDEs can have solutionswith discontinuities. This paper investigates, both theoretically andempirically, the operator learning of PDEs with discontinuous solutions. Werigorously prove, in terms of lower approximation bounds, that methods whichentail a linear reconstruction step (e.g. DeepONet or PCA-Net) fail toefficiently approximate the solution operator of such PDEs. In contrast, weshow that certain methods employing a non-linear reconstruction mechanism canovercome these fundamental lower bounds and approximate the underlying operatorefficiently. The latter class includes Fourier Neural Operators and a novelextension of DeepONet termed shift-DeepONet. Our theoretical findings areconfirmed by empirical results for advection equation, inviscid Burgers'equation and compressible Euler equations of aerodynamics.

Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities