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

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.