Backward Reconstruction of the Chafee--Infante Equation via Physics-Informed WGAN-GP
Joseph L. Shomberg
TL;DR
This work tackles the ill-posed inverse problem of reconstructing an initial condition for the bistable Chafee--Infante equation from a near-equilibrium snapshot. It introduces a physics-informed WGAN-GP with a forward-simulation residual, trained on data generated by forward Euler dynamics, and augmented with Lyapunov energy matching and moment statistics to enforce physical and statistical consistency. The key contribution is the residual term computed with the same discretization used to generate data, ensuring discrete dynamical admissibility and enabling sharp-interface recovery; the method achieves a test MAE of about $0.2399\pm0.0027$ on 10k samples, demonstrating stable inversion and robustness to high-frequency noise. This numerically aligned physics-informed adversarial framework offers a practical pathway for solving strongly dissipative inverse problems where traditional regularization or PINN-based methods struggle, with potential applicability to other reaction-diffusion systems.
Abstract
We present a physics-informed Wasserstein GAN with gradient penalty (WGAN-GP) for solving the inverse Chafee--Infante problem on two-dimensional domains with Dirichlet boundary conditions. The objective is to reconstruct an unknown initial condition from a near-equilibrium state obtained after 100 explicit forward Euler iterations of the reaction-diffusion equation \[ u_t - γΔu + κ\left(u^3 - u\right)=0. \] Because this mapping strongly damps high-frequency content, the inverse problem is severely ill-posed and sensitive to noise. Our approach integrates a U-Net generator, a PatchGAN critic with spectral normalization, Wasserstein loss with gradient penalty, and several physics-informed auxiliary terms, including Lyapunov energy matching, distributional statistics, and a crucial forward-simulation penalty. This penalty enforces consistency between the predicted initial condition and its forward evolution under the \emph{same} forward Euler discretization used for dataset generation. Earlier experiments employing an Eyre-type semi-implicit solver were not compatible with this residual mechanism due to the cost and instability of Newton iterations within batched GPU training. On a dataset of 50k training and 10k testing pairs on $128\times128$ grids (with natural $[-1,1]$ amplitude scaling), the best trained model attains a mean absolute error (MAE) of approximately \textbf{0.23988159} on the full test set, with a sample-wise standard deviation of about \textbf{0.00266345}. The results demonstrate stable inversion, accurate recovery of interfacial structure, and robustness to high-frequency noise in the initial data.