Convergence of physics-informed neural networks for time-fractional diffusion equations via stability and generalization bounds

Phuoc Toan Trinh; Huu Dinh Huynh

doi:10.61383/ejam.202534116

Authors

Trinh Phuoc Toan Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Vietnam https://orcid.org/0000-0002-7707-322X
Huynh Huuu Dinh Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Vietnam Corresponding Author https://orcid.org/0000-0001-9762-0203

DOI:

https://doi.org/10.61383/ejam.202534116

Keywords:

Physics-informed neural networks, time-fractional diffusion, Caputo derivative, stability estimate, generalization error, convergence

Abstract

Physics-informed neural networks (PINNs) have emerged as a flexible framework for solving partial differential equations (PDEs), yet rigorous convergence results remain limited for models with nonlocal time dependence. This paper studies PINNs for the time-fractional diffusion equation with the Caputo derivative of order \(\alpha\in(0,1)\). We propose a residual-based PINN formulation and establish a stability-driven error bound: the solution error is controlled by the PDE residual together with the boundary and initial condition residuals. By combining this PDE stability with sampling-based generalization bounds for the empirical PINN loss, we prove convergence of empirical minimizers to the true solution as the number of collocation points increases and the approximation and optimization errors vanish. Numerical experiments with manufactured solutions support the theory and demonstrate accuracy.

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