In this paper, we propose the augmented physics-informed neural network (APINN), which adopts soft and trainable domain decomposition and flexible parameter sharing to further improve the extended PINN (XPINN) as well as the vanilla PINN methods. In particular, a trainable gate network is employed to mimic the hard decomposition of XPINN, which can be flexibly fine-tuned for discovering a potentially better partition. It weight-averages several sub-nets as the output of APINN. APINN does not require complex interface conditions, and its sub-nets can take advantage of all training samples rather than just part of the training data in their subdomains. Lastly, each sub-net shares part of the common parameters to capture the similar components in each decomposed function. Furthermore, following the PINN generalization theory in Hu et al. [2021], we show that APINN can improve generalization by proper gate network initialization and general domain & function decomposition. Extensive experiments on different types of PDEs demonstrate how APINN improves the PINN and XPINN methods. Specifically, we present examples where XPINN performs similarly to or worse than PINN, so that APINN can significantly improve both. We also show cases where XPINN is already better than PINN, so APINN can still slightly improve XPINN. Furthermore, we visualize the optimized gating networks and their optimization trajectories, and connect them with their performance, which helps discover the possibly optimal decomposition. Interestingly, if initialized by different decomposition, the performances of corresponding APINNs can differ drastically. This, in turn, shows the potential to design an optimal domain decomposition for the differential equation problem under consideration.

翻译：本文提出了增强型物理信息神经网络（APINN），该网络采用软且可训练的区域分解与灵活的参数共享机制，以进一步改进扩展型PINN（XPINN）及原始PINN方法。具体而言，我们使用可训练的门控网络模拟XPINN的硬区域分解，通过灵活微调以发现潜在更优的划分方案。APINN以加权平均方式组合多个子网络作为最终输出。该方法无需复杂的界面条件，且其子网络能够利用所有训练样本（而非仅局限于子区域内的部分数据）。最后，每个子网络共享部分公共参数以捕捉各分解函数中的相似成分。此外，基于Hu等人[2021]提出的PINN泛化理论，我们证明通过合理的门控网络初始化与通用区域和函数分解，APINN可提升泛化性能。在多种偏微分方程上的大量实验展示了APINN如何改进PINN与XPINN方法。具体而言，我们展示了XPINN性能与PINN相当或更差的案例，此时APINN可显著提升二者性能；同时展示XPINN已优于PINN的案例，此时APINN仍能小幅改进XPINN。我们还可视化了优化后的门控网络及其优化轨迹，并将其与性能关联，从而有助于发现潜在最优分解。有趣的是，若采用不同分解初始化，对应APINN的性能差异显著。这反过来表明针对所考虑的微分方程问题设计最优区域分解的可能性。