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.

翻译：本文提出一种基于物理信息的带梯度惩罚Wasserstein生成对抗网络（WGAN-GP），用于求解具有Dirichlet边界条件的二维区域上的Chafee–Infante逆向问题。目标是从反应-扩散方程经过100次显式前向欧拉迭代后获得的近平衡状态中重构未知的初始条件，该方程形式为\[ u_t - γΔu + κ\left(u^3 - u\right)=0. \]由于该映射强烈抑制高频成分，该逆向问题具有严重的不适定性且对噪声敏感。我们的方法整合了U-Net生成器、采用谱归一化的PatchGAN判别器、带梯度惩罚的Wasserstein损失函数，以及多项物理信息辅助项，包括Lyapunov能量匹配、分布统计量，以及关键的前向模拟惩罚项。该惩罚项通过**相同**于数据集生成所用的前向欧拉离散格式，强制保证预测初始条件与其前向演化之间的一致性。早期采用Eyre型半隐式求解器的实验由于在批处理GPU训练中牛顿迭代的成本与不稳定性，无法与此残差机制兼容。在$128\times128$网格（采用自然$[-1,1]$幅值缩放）上包含50k训练对和10k测试对的数据集上，最优训练模型在整个测试集上达到约**0.23988159**的平均绝对误差（MAE），样本级标准差约为**0.00266345**。结果表明该方法能实现稳定的逆向求解、精确恢复界面结构，并对初始数据中的高频噪声具有鲁棒性。