# 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.

## Key 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.