The Finite Element Neural Network Method: One Dimensional Study

TL;DR

This paper introduces the finite element neural network method (FENNM), combining neural networks with finite element principles to improve accuracy and applicability in solving differential equations, demonstrated through numerical case studies.

Contribution

The paper presents FENNM, a novel approach integrating neural networks with finite element methods within the Petrov-Galerkin framework, enhancing solution accuracy and extending applicability.

Findings

FENNM incorporates flux terms into the loss function, improving solution quality.

The method enables integration of forcing and boundary conditions similar to traditional FEM.

Numerical case studies demonstrate FENNM's robustness and accuracy.

Abstract

The potential of neural networks (NN) in engineering is rooted in their capacity to understand intricate patterns and complex systems, leveraging their universal nonlinear approximation capabilities and high expressivity. Meanwhile, conventional numerical methods, backed by years of meticulous refinement, continue to be the standard for accuracy and dependability. Bridging these paradigms, this research introduces the finite element neural network method (FENNM) within the framework of the Petrov-Galerkin method using convolution operations to approximate the weighted residual of the differential equations. The NN generates the global trial solution, while the test functions belong to the Lagrange test function space. FENNM introduces several key advantages. Notably, the weak-form of the differential equations introduces flux terms that contribute information to the loss function compared to VPINN, hp-VPINN, and cv-PINN. This enables the integration of forcing terms and natural boundary conditions into the loss function similar to conventional finite element method (FEM) solvers, facilitating its optimization, and extending its applicability to more complex problems, which will ease industrial adoption. This study will elaborate on the derivation of FENNM, highlighting its similarities with FEM. Additionally, it will provide insights into optimal utilization strategies and user guidelines to ensure cost-efficiency. Finally, the study illustrates the robustness and accuracy of FENNM by presenting multiple numerical case studies and applying adaptive mesh refinement techniques.