An h-adaptive collocation method for Physics-Informed Neural Networks

• Published in: Journal of computational science Vol. 91; p. 102684
• Main Authors: Trynda, Jan, Maczuga, Paweł, Oliver-Serra, Albert, García-Castillo, Luis Emilio, Schaefer, Robert, Woźniak, Maciej
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2025
• Subjects: Advection-dominated problems; Almost-singular Poisson problems; Physics-Informed Neural Networks; Residual-based adaptive sampling; Physics-Informed Neural Networks; Residual-based adaptive sampling; Advection-dominated problems; Almost-singular Poisson problems
• ISSN: 1877-7503
• DOI: 10.1016/j.jocs.2025.102684

Despite their flexibility and success in solving partial differential equations, Physics-Informed Neural Networks (PINNs) often suffer from convergence issues, even failing to converge, particularly in problems with steep gradients or localized features. Several remedies have been suggested to solve this problem, but one of the most promising is the dynamical adaptation of the collocation points. This paper explores a novel adaptive sampling method, of a stochastic nature, based on the Adaptive Mesh Refinement used in the Finite Element Method. The error estimates in our refinement algorithm are based on the value of the residual loss function. We tested our method against a variety of 1D and 2D benchmark problems that exhibit steep gradients near certain boundaries, with promising results. •Adaptive sampling for PINNs using residual-based mesh refinement.•Captures steep gradients and boundary layers for faster convergence.•Works for 1D and 2D PDEs, including advection-dominated and elliptic cases.•Maintains flexibility for higher dimensions and integrates with other methods.