Adversarial Training for Physics-Informed Neural Networks
Yao Li Shengzhu Shi Zhichang Guo Boying Wu
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.