Physics-informed neural networks have shown great promise in solving partial differential equations. However, due to insufficient robustness, vanilla PINNs often face challenges when solving complex PDEs, especially those involving multi-scale behaviors or solutions with sharp or oscillatory characteristics. To address these issues, based on the projected gradient descent adversarial attack, we proposed an adversarial training strategy for PINNs termed by AT-PINNs. AT-PINNs enhance the robustness of PINNs by fine-tuning the model with adversarial samples, which can accurately identify model failure locations and drive the model to focus on those regions during training. AT-PINNs can also perform inference with temporal causality by selecting the initial collocation points around temporal initial values. We implement AT-PINNs to the elliptic equation with multi-scale coefficients, Poisson equation with multi-peak solutions, Burgers equation with sharp solutions and the Allen-Cahn equation. The results demonstrate that AT-PINNs can effectively locate and reduce failure regions. Moreover, AT-PINNs are suitable for solving complex PDEs, since locating failure regions through adversarial attacks is independent of the size of failure regions or the complexity of the distribution.

翻译：物理信息神经网络在求解偏微分方程方面展现出巨大潜力。然而，由于鲁棒性不足，基础PINNs在求解复杂PDEs时常常面临挑战，尤其是涉及多尺度行为或具有陡峭/振荡特征的解时。针对这些问题，基于投影梯度下降对抗攻击方法，我们提出了一种面向PINNs的对抗训练策略，称为AT-PINNs。AT-PINNs通过使用对抗样本对模型进行微调来增强鲁棒性，这些样本能准确识别模型失效区域，并驱动模型在训练过程中聚焦于这些区域。AT-PINNs还可通过选取时间初始值附近的初始配置点实现时间因果推理。我们将AT-PINNs应用于多尺度系数椭圆方程、多峰解泊松方程、具有陡峭解的Burgers方程以及Allen-Cahn方程。结果表明，AT-PINNs能有效定位并缩小失效区域。此外，由于通过对抗攻击定位失效区域与失效区域大小或分布复杂度无关，AT-PINNs适用于求解复杂PDEs。