Convolutional Neural Operators for robust and accurate learning of PDEs

TL;DR

This paper introduces convolutional neural operators (CNOs), a novel neural network architecture that effectively learns solution operators of PDEs, demonstrating universality and superior performance on diverse benchmarks.

Contribution

The paper adapts convolutional neural networks for operator learning, proving their universality and demonstrating their effectiveness for PDEs, which was previously unexplored.

Findings

CNOs can approximate PDE operators with high accuracy

CNOs outperform existing methods on diverse PDE benchmarks

CNOs preserve continuous properties even in discretized implementations

Abstract

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with possibly multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning. Our code is publicly available at https://github.com/bogdanraonic3/ConvolutionalNeuralOperator