Backward Reconstruction of the Chafee--Infante Equation via Physics-Informed WGAN-GP

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.