Symmetries have been leveraged to improve the generalization of neural
networks through different mechanisms from data augmentation to equivariant
architectures. However, despite their potential, their integration into neural
solvers for partial differential equations (PDEs) remains largely unexplored.
We explore the integration of PDE symmetries, known as Lie point symmetries, in
a major family of neural solvers known as physics-informed neural networks
(PINNs). We propose a loss function that informs the network about Lie point
symmetries in the same way that PINN models try to enforce the underlying PDE
through a loss function. Intuitively, our symmetry loss ensures that the
infinitesimal generators of the Lie group conserve the PDE solutions.
Effectively, this means that once the network learns a solution, it also learns
the neighbouring solutions generated by Lie point symmetries. Empirical
evaluations indicate that the inductive bias introduced by the Lie point
symmetries of the PDEs greatly boosts the sample efficiency of PINNs.

我们研究了将偏微分方程 (PDE) 的对称性，即李点对称性，整合到物理信息神经网络 (PINNs) 中的方法，并提出了一种损失函数，用于告知网络关于李点对称性，从而提高了 PINNs 的样本效率。