Hybrid FEM-RBFNN: A Fusion of Finite Element Method and Radial Basis Function Neural Networks for Solving PDEs

Ojo, Ridwan O; Adebisi, Ajimot; Tajudeen, Ganiyu; Ogunniran, Muideen O; Bayram, M.; Kareem, K.O.

doi:10.30511/ttmp.2026.2087762.1074

Hybrid FEM-RBFNN: A Fusion of Finite Element Method and Radial Basis Function Neural Networks for Solving PDEs

Articles in Press

Document Type : Original Article

Authors

Ridwan O Ojo ¹

Ajimot Adebisi ¹

Ganiyu Tajudeen ¹

Muideen O Ogunniran ¹

M. Bayram ²

K.O. Kareem ³

¹ Department of Mathematical Sciences, Osun State University, Osogbo, Nigeria

² Faculty of Engineering, Biruni University, Turkey

³ Department of Mathematical Sciences, Federal College of Education, Iwo, Nigeria

10.30511/ttmp.2026.2087762.1074

Abstract

The Finite Element Method (FEM) serves as a standard numerical technique for Partial Differential Equation (PDE) solutions because it maintains stability, combined with theoretical proof. The methods face major performance issues when dealing with high-dimensional spaces that need detailed mesh resolutions. The research introduces a hybrid FEM-RBFNN framework, which functions as a fast substitute model to study nonlinear wave movements. The hybrid method uses Radial Basis Function Neural Network (RBFNN) training on low-cost coarse-grid FEM data to produce accurate results without needing full mesh refinement. The nonlinear Schrödinger Equation (NLSE) analysis shows that FEM-RBFNN successfully tracks complex soliton movement while it reduces numerical problems that arise from using coarse mesh grids. The results demonstrate that the system achieves a 60x speed increase over detailed ground truth simulations, which decreases processing time from 4.60s to 0.08s. The surrogate maintains an L2 error of 0.014, which matches the performance of the coarse source while it uses training data to solve high-frequency dispersive tails. The research develops an efficient PDE solver which handles real-time simulations and extensive parameter research through its robust and differentiable features.

Keywords

Finite Element, Nonlinear Schrö

dinger Equation, Radial Basis Function, Neural Networks

Subjects