BelNet: Basis enhanced learning, a mesh-free neural operator

TL;DR

BelNet introduces a mesh-free neural operator that learns basis functions to efficiently solve parametric PDEs, enabling flexible discretization and improved performance on complex multiscale problems.

Contribution

The paper presents BelNet, a novel mesh-free neural operator that learns basis functions, extending operator learning to handle different input/output meshes and discretizations.

Findings

Outperforms existing operator learning methods on high-contrast problems

Allows flexible sampling and discretization of input/output functions

Demonstrates effectiveness on multiscale PDEs

Abstract

Operator learning trains a neural network to map functions to functions. An ideal operator learning framework should be mesh-free in the sense that the training does not require a particular choice of discretization for the input functions, allows for the input and output functions to be on different domains, and is able to have different grids between samples. We propose a mesh-free neural operator for solving parametric partial differential equations. The basis enhanced learning network (BelNet) projects the input function into a latent space and reconstructs the output functions. In particular, we construct part of the network to learn the ``basis'' functions in the training process. This generalized the networks proposed in Chen and Chen's universal approximation theory for the nonlinear operators to account for differences in input and output meshes. Through several challenging high-contrast and multiscale problems, we show that our approach outperforms other operator learning methods for these tasks and allows for more freedom in the sampling and/or discretization process.