Physics-informed neural networks and operator networks have shown promise for effectively solving equations modeling physical systems. However, these networks can be difficult or impossible to train accurately for some systems of equations. We present a novel multifidelity framework for stacking physics-informed neural networks and operator networks that facilitates training. We successively build a chain of networks, where the output at one step can act as a low-fidelity input for training the next step, gradually increasing the expressivity of the learned model. The equations imposed at each step of the iterative process can be the same or different (akin to simulated annealing). The iterative (stacking) nature of the proposed method allows us to progressively learn features of a solution that are hard to learn directly. Through benchmark problems including a nonlinear pendulum, the wave equation, and the viscous Burgers equation, we show how stacking can be used to improve the accuracy and reduce the required size of physics-informed neural networks and operator networks.

翻译：物理信息神经网络与算子网络在有效求解物理系统建模方程方面展现出潜力。然而，对于某些方程组，这些网络可能难以或无法实现精确训练。我们提出一种新颖的多保真度框架，通过堆叠物理信息神经网络与算子网络来促进训练过程。该框架通过逐步构建网络链，其中上一步的输出可作为下一步训练的低保真度输入，从而逐步增强所学习模型的表达能力。迭代过程中各步骤施加的方程可以相同或不同（类似于模拟退火方法）。所提方法的迭代（堆叠）特性使我们能够渐进学习难以直接求解的特征。通过非线性单摆、波动方程及粘性伯格斯方程等基准问题，我们验证了堆叠方法可有效提升物理信息神经网络与算子网络的精度，并减少所需网络规模。