A discretization-invariant extension and analysis of some deep operator networks

TL;DR

This paper introduces a generalized discretization-invariant neural operator that universally approximates operators, allowing flexible sampling of input functions across different discretizations, with demonstrated superior performance on multiscale PDEs.

Contribution

It extends neural operators to be discretization-invariant and establishes their universality, connecting them with previous models and enabling flexible input sampling.

Findings

Achieves lower prediction errors than previous networks.

Demonstrates discretization-invariance in multiscale PDE examples.

Provides a theoretical foundation for universal approximation of operators.

Abstract

We present a generalized version of the discretization-invariant neural operator and prove that the network is a universal approximation in the operator sense. Moreover, by incorporating additional terms in the architecture, we establish a connection between this discretization-invariant neural operator network and those discussed before. The discretization-invariance property of the operator network implies that different input functions can be sampled using various sensor locations within the same training and testing phases. Additionally, since the network learns a ``basis'' for the input and output function spaces, our approach enables the evaluation of input functions on different discretizations. To evaluate the performance of the proposed discretization-invariant neural operator, we focus on challenging examples from multiscale partial differential equations. Our experimental results indicate that the method achieves lower prediction errors compared to previous networks and benefits from its discretization-invariant property.