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 are 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.

翻译：多尺度问题对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）构成挑战，这（部分）可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间依赖问题中的性能，本文结合了多保真度堆叠物理信息神经网络与基于区域分解的有限基物理信息神经网络。具体而言，为学习多保真度模型的高保真度部分，采用了时间上的区域分解方法。针对单摆问题、双频率问题以及艾伦-卡恩方程，本文研究了该方法的性能表现。可以观察到，区域分解方法显著改进了物理信息神经网络及堆叠物理信息神经网络的方法。