论文标题
非线性重建用于操作员学习不连续性的PDE
Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities
论文作者
论文摘要
一大批双曲线和以平流为主的PDE可以具有不连续性的解决方案。本文从理论上和经验上都研究了操作员以不连续解决方案学习PDE。我们严格地证明，从较低的近似边界角度来看，需要线性重建步骤（例如deeponet或pca-net）的方法无法有效地近似于此类PDES的解决方案操作员。相比之下，我们表明，采用非线性重建机制的某些方法可以克服这些基本的下限，并有效地近似基础操作员。后一类包括傅立叶神经操作员和被称为Shift-Deeponet的DeepOnet的新型扩展。我们的理论发现通过对流方程，无粘性汉堡方程和可压缩的欧拉动力学方程的经验结果证实。
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.
