Multiscale problems are challenging for neural network-based discretizations of differential equations, such as physics-informed neural networks (PINNs). This can be (partly) attributed to the so-called spectral bias of neural networks. To improve the performance of PINNs for time-dependent problems, a combination of multifidelity stacking PINNs and domain decomposition-based finite basis PINNs is employed. In particular, to learn the high-fidelity part of the multifidelity model, a domain decomposition in time is employed. The performance is investigated for a pendulum and a two-frequency problem as well as the Allen-Cahn equation. It can be observed that the domain decomposition approach clearly improves the PINN and stacking PINN approaches. Finally, it is demonstrated that the FBPINN approach can be extended to multifidelity physics-informed deep operator networks.

翻译：多尺度问题对于基于神经网络的微分方程离散化方法（如物理信息神经网络）具有挑战性。这可以（部分地）归因于神经网络所谓的谱偏差。为提升物理信息神经网络在时变问题上的性能，本文采用了多保真度堆叠物理信息神经网络与基于域分解的有限基物理信息神经网络的组合方法。具体而言，为学习多保真度模型的高保真部分，采用了时间域分解策略。研究通过摆锤问题、双频率问题以及Allen-Cahn方程验证了该方法的性能。可以观察到，域分解方法显著改进了传统物理信息神经网络及堆叠物理信息神经网络方法。最后，研究证明了有限基物理信息神经网络方法可扩展至多保真度物理信息深度算子网络。