WF-PINNs: solving forward and inverse problems of burgers equation with steep gradients using weak-form physics-informed neural networks

TL;DR

The paper introduces a new neural network framework to solve complex fluid dynamics problems involving shocks with improved accuracy and stability.

Contribution

The novel weak-form PINN framework enables stable and accurate solutions for forward and inverse problems with steep gradients in conservation laws.

Findings

WF-PINNs outperform strong-form PINNs in accuracy and robustness for problems with shocks.

The framework successfully identifies unknown initial conditions and viscosity coefficients in inverse problems.

The weak-form formulation and entropy condition improve stability and physical consistency near shocks.

Abstract

This study tackles the numerical challenges posed by solutions with steep gradients in the Burgers equation, particularly poor stability in high-gradient regions and the ill-posedness of inverse problems in shock wave modeling. We propose a Weak-Form Physics-Informed Neural Network (WF-PINN) that fundamentally enhances both forward and inverse problem solving. Key innovations include: (i) a weak-form integral formulation of the PDE loss, which improves training stability near shocks; (ii) enforcement of an entropy condition to ensure unique and physically consistent shock capture; (iii) a dual-network architecture for inverse problems, where an auxiliary network dedicated to initial condition reconstruction is coupled with the main solver via consistency constraints. Numerical experiments show that WF-PINNs achieve significantly higher accuracy and convergence robustness compared to strong-form PINNs, accurately resolving shock locations and amplitudes while enabling precise identification of unknown initial conditions and viscosity coefficients. The framework offers a unified and generalizable approach for solving conservation laws with discontinuities.

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