The Finite Element Neural Network Method: One Dimensional Study

TL;DR

The paper introduces the Finite Element Neural Network Method (FENNM), a Petrov-Galerkin-inspired neural solver that embeds flux information from the weak form into a convolution-based residual, using a neural network to represent the global trial space and nonvanishing Lagrange test functions to allow natural and intermediate boundary terms. By leveraging convolution for Gauss quadrature and enforcing boundary conditions within the residual loss, FENNM achieves FEM-like accuracy with ML flexibility, demonstrated across one-dimensional benchmark problems including beam bending, a nonlinear pendulum, transport, and Poisson equations with steep or discontinuous features. The results show favorable convergence behavior, effective adaptive mesh refinement, and competitive rate of convergence compared to classical FEM, while maintaining a relatively fixed network DOF and enabling time-like extensions. Overall, FENNM narrows the gap between ML-based solvers and traditional numerical methods, offering a scalable path toward higher-dimensional and parameterized problems with improved optimization and practical industrial applicability.

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.

Figures (12)

• Figure 1: Schematic of one-dimensional FENNM. The NN outputs formulate the fluxes, differential equation operators, and forcing terms. Then predefined filters pass over them to compute the Gauss quadrature sums in the convolution process before evaluating the global residual loss function.
• Figure 3: An illustration of a uniform mesh comprising five elements over the domain $x \in [1,2]$. Markers indicate: the quadrature points $x_q$ distributed throughout the elements in the global coordinates; left boundary condition; right boundary condition; elements' right boundaries; elements' left boundaries.
• Figure 4: The average absolute error $\mathcal{E}(x) = 1/N\sum_{i=1}^{N}|U_{NN}(x)-U(x)|$ in FENNM when increasing the number of the quadrature points for different test function orders.
• Figure 5: The rate of convergence of FENNM using linear, quadratic, and cubic test functions at $x=1.5$. The error bars represent 95% confidence interval, calculated from ten randomly initialized networks for each mesh.
• Figure 6: Solution of a cantilevered beam subjected to an intermediate static force using FENNM: (a) FENNM approximation compared to the exact solution; (b) PWE over the beam length; (c) training history of the FENNM. The beam is of length $L=1 [m]$, modulus of elasticity $E=69.9\times 10^9 [GPa]$, second moment of area of $I=9\times 10^{-9}[m^4]$, and a point load $F=100 [N]$.