Multi-level neural networks for PDEs with uncertain parameters

TL;DR

This paper introduces a multi-level neural network approach for solving PDEs with uncertain parameters, leveraging convolutional neural networks and transfer learning to improve accuracy and efficiency over existing methods.

Contribution

The paper presents a novel multi-level neural network framework that captures error structures in PDE solutions and reuses information across grid levels, enhancing performance for complex and high-accuracy problems.

Findings

Outperforms state-of-the-art multi-level methods

Effective for complex PDEs like free-surface flow

Reduces sampling needs on fine grid levels

Abstract

A novel multi-level method for partial differential equations with uncertain parameters is proposed. The principle behind the method is that the error between grid levels in multi-level methods has a spatial structure that is by good approximation independent of the actual grid level. Our method learns this structure by employing a sequence of convolutional neural networks, that are well-suited to automatically detect local error features as latent quantities of the solution. Furthermore, by using the concept of transfer learning, the information of coarse grid levels is reused on fine grid levels in order to minimize the required number of samples on fine levels. The method outperforms state-of-the-art multi-level methods, especially in the case when complex PDEs (such as single-phase and free-surface flow problems) are concerned, or when high accuracy is required.