Mesh-Informed Neural Networks for Operator Learning in Finite Element Spaces

TL;DR

This paper introduces Mesh-Informed Neural Networks (MINNs), a novel architecture tailored for mesh-based functional data, demonstrating improved efficiency and generalization in operator learning for PDEs compared to traditional neural network models.

Contribution

The paper presents MINNs, a new neural network architecture that embeds layers into discrete functional spaces, enabling sparse, efficient learning of nonlinear operators on complex domains.

Findings

MINNs outperform traditional architectures in training efficiency and generalization.

MINNs handle functional data on arbitrary domain shapes effectively.

MINNs are well-suited for reduced order modeling and uncertainty quantification.

Abstract

Thanks to their universal approximation properties and new efficient training strategies, Deep Neural Networks are becoming a valuable tool for the approximation of mathematical operators. In the present work, we introduce Mesh-Informed Neural Networks (MINNs), a class of architectures specifically tailored to handle mesh based functional data, and thus of particular interest for reduced order modeling of parametrized Partial Differential Equations (PDEs). The driving idea behind MINNs is to embed hidden layers into discrete functional spaces of increasing complexity, obtained through a sequence of meshes defined over the underlying spatial domain. The approach leads to a natural pruning strategy which enables the design of sparse architectures that are able to learn general nonlinear operators. We assess this strategy through an extensive set of numerical experiments, ranging from nonlocal operators to nonlinear diffusion PDEs, where MINNs are compared against more traditional architectures, such as classical fully connected Deep Neural Networks, but also more recent ones, such as DeepONets and Fourier Neural Operators. Our results show that MINNs can handle functional data defined on general domains of any shape, while ensuring reduced training times, lower computational costs, and better generalization capabilities, thus making MINNs very well-suited for demanding applications such as Reduced Order Modeling and Uncertainty Quantification for PDEs.