Variationally Mimetic Operator Networks

TL;DR

This paper introduces a variationally mimetic operator network (VarMiON) architecture that improves PDE solution approximation by mimicking variational formulations, showing enhanced accuracy and robustness over existing operator networks.

Contribution

The paper proposes a new VarMiON architecture with a precisely determined structure, improving PDE approximation accuracy and robustness compared to DeepONet and MIONet.

Findings

VarMiON achieves smaller errors than DeepONet and MIONet for similar network sizes.

VarMiON demonstrates greater robustness to input variations and sampling techniques.

Error analysis reveals contributions from training data, quadrature, and covering errors.

Abstract

In recent years operator networks have emerged as promising deep learning tools for approximating the solution to partial differential equations (PDEs). These networks map input functions that describe material properties, forcing functions and boundary data to the solution of a PDE. This work describes a new architecture for operator networks that mimics the form of the numerical solution obtained from an approximate variational or weak formulation of the problem. The application of these ideas to a generic elliptic PDE leads to a variationally mimetic operator network (VarMiON). Like the conventional Deep Operator Network (DeepONet) the VarMiON is also composed of a sub-network that constructs the basis functions for the output and another that constructs the coefficients for these basis functions. However, in contrast to the DeepONet, the architecture of these sub-networks in the VarMiON is precisely determined. An analysis of the error in the VarMiON solution reveals that it contains contributions from the error in the training data, the training error, the quadrature error in sampling input and output functions, and a "covering error" that measures the distance between the test input functions and the nearest functions in the training dataset. It also depends on the stability constants for the exact solution operator and its VarMiON approximation. The application of the VarMiON to a canonical elliptic PDE and a nonlinear PDE reveals that for approximately the same number of network parameters, on average the VarMiON incurs smaller errors than a standard DeepONet and a recently proposed multiple-input operator network (MIONet). Further, its performance is more robust to variations in input functions, the techniques used to sample the input and output functions, the techniques used to construct the basis functions, and the number of input functions.