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Date
2022-10Type
- Report
ETH Bibliography
yes
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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. Show more
Publication status
publishedExternal links
Journal / series
SAM Research ReportVolume
Publisher
Seminar for Applied Mathematics, ETH ZurichOrganisational unit
03851 - Mishra, Siddhartha / Mishra, Siddhartha
Funding
770880 - Computation and analysis of statistical solutions of fluid flow (EC)
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ETH Bibliography
yes
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