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Physics Informed Neural Networks with strong and weak residuals for advection-dominated diffusion problems

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arxiv 2307.07647 v1 pith:DV447KWM submitted 2023-07-14 math.NA cs.NA

Maciej Sikora , Patryk Krukowski , Anna Paszynska , Maciej Paszynski This is my paper · ORCID
classification math.NAcs.NA
keywords pinnvpinnadvection-dominateddiffusionproblemfunctionsinformedmethod
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This paper deals with the following important research questions. Is it possible to solve challenging advection-dominated diffusion problems in one and two dimensions using Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN)? How does it compare to the higher-order and continuity Finite Element Method (FEM)? How to define the loss functions for PINN and VPINN so they converge to the correct solutions? How to select points or test functions for training of PINN and VPINN? We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson-Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov-Galerkin (PG) formulation and present the FEM solution obtained with the PG method. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems

    math.AP 2026-07 conditional novelty 6.0 of 10

    Integrated Cauchy ridge atoms give thickness-uniform, derivative-stable approximation of analytic convection layers and yield more accurate, parameter-efficient PINNs for singularly perturbed convection–diffusion problems.

  2. Physical Informed Neural Networks for modeling ocean pollutant

    cs.LG 2025-07 conditional novelty 4.0 of 10

    A physics-informed neural network trained in Julia fits noisy finite-difference data for a constant-velocity 2D advection-diffusion problem and reports an 8.25% relative L2 error against the same solver's clean output.

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