Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities
METADATA ONLY
Loading...
Author / Producer
Date
2022-10
Publication Type
Report
ETH Bibliography
yes
Citations
Altmetric
METADATA ONLY
Data
Rights / License
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.
Permanent link
Publication status
published
Editor
Book title
Journal / series
Volume
2022-42
Pages / Article No.
Publisher
Seminar for Applied Mathematics, ETH Zurich
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Organisational unit
03851 - Mishra, Siddhartha / Mishra, Siddhartha
Notes
Funding
770880 - Computation and analysis of statistical solutions of fluid flow (EC)