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.

基于梯度下降对抗攻击，本文提出了一种适用于物理知识的神经网络（PINNs）的对抗训练策略，称为 AT-PINNs，通过对抗样本微调模型，从而提高了 PINNs 的鲁棒性，准确识别模型失败区域，并在训练过程中使模型关注这些区域。AT-PINNs 还能通过选择初始值周围的对应点进行推理，从而执行具有时间因果关系的推理。实验结果表明，AT-PINNs 能够有效地定位和减少失败区域，且适用于解决复杂的偏微分方程，因为通过对抗攻击定位失败区域与失败区域的大小或分布的复杂程度无关。