Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities
This addresses a critical bottleneck in machine learning for physics simulations, particularly in aerodynamics and hyperbolic PDEs, by providing theoretical and empirical evidence for the necessity of nonlinear reconstruction, representing a significant advance over incremental improvements.
The paper tackled the problem of operator learning for PDEs with discontinuous solutions, proving that linear reconstruction methods like DeepONet fail to efficiently approximate such operators, while nonlinear methods like Fourier Neural Operators and a novel shift-DeepONet can overcome these bounds and achieve efficient approximation, as confirmed by empirical tests on advection, Burgers', and Euler equations.
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 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 DeepONet termed shift-DeepONet. Our theoretical findings are confirmed by empirical results for advection equation, inviscid Burgers' equation and compressible Euler equations of aerodynamics.