Numerically Informed Convolutional Operator Network with Subproblem Decomposition for Poisson Equations

TL;DR

This paper introduces NICON, a neural operator that integrates classical numerical methods with deep learning, providing theoretical error estimates and training strategies for solving Poisson equations efficiently on refined grids.

Contribution

The work presents a novel numerically informed convolutional operator network that couples finite difference and finite element methods with neural operator learning, including error analysis and convergence guarantees.

Findings

Error estimates relate convergence to loss decay rate.

Training strategies guarantee optimal convergence under grid refinement.

Numerical experiments validate theoretical predictions and performance.

Abstract

Neural operators have shown remarkable performance in approximating solutions of partial differential equations. However, their convergence behavior under grid refinement is still not well understood from the viewpoint of numerical analysis. In this work, we propose a numerically informed convolutional operator network, called NICON, that explicitly couples classical finite difference and finite element methods with operator learning through residual-based training loss functions. We introduce two types of networks, FD-CON and FE-CON, which use residual-based loss functions derived from the corresponding numerical methods. We derive error estimates for FD-CON and FE-CON using finite difference and finite element analysis. These estimates show a direct relation between the convergence behavior and the decay rate of the training loss. From these analyses, we establish training strategies that guarantee optimal convergence rates under grid refinement. Several numerical experiments are presented to validate the theoretical results and show performance on fine grids.