A Constrained FEM–LSSVR Framework with B-Spline Basis for Data-Driven Solution of Nonlinear PDEs

Articles in Press

Document Type : Research article

Authors

1 Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

2 Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran; Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, Iran

Abstract

This paper presents a constrained hybrid FEM-LSSVR framework for the numerical solution of nonlinear partial differential equations, with a particular focus on FitzHugh-Nagumo (FHN)-type model variants. In the proposed approach, Least Squares Support Vector Regression (LSSVR) is not employed as a post-processing or data-fitting tool; instead, it is formulated as a local primal optimization problem in which the governing PDE, boundary conditions, and inter-element continuity are explicitly enforced as constraints. The finite element method (FEM) is incorporated to provide structural consistency, embedding boundary and continuity information directly into the learning formulation rather than using FEM solutions as training data. The resulting method establishes a tight interaction between numerical discretization and constrained learning, yielding a flexible yet physically consistent approximation framework. The performance of the proposed approach is investigated through numerical experiments on several FHN-type nonlinear PDE variants, including studies with varying mesh resolutions, different final time horizons, and data-driven settings. Comparisons with standard FEM solutions demonstrate improved robustness and accuracy, particularly in scenarios involving limited or noisy data. These results indicate that the constrained FEM-LSSVR formulation provides a viable alternative to traditional numerical solvers for nonlinear PDEs in data-driven contexts.

Keywords

Main Subjects

Analytical and Numerical Solutions for Nonlinear Equations

Articles in Press, Accepted Manuscript
Available Online from 05 October 2026

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History

  • Receive Date: 25 August 2026
  • Revise Date: 08 September 2026
  • Accept Date: 19 September 2026
  • Publish Date: 05 October 2026

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