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.

通过构建一系列的网络，其中一个步骤的输出可以作为下一步训练的低保真输入，我们提出了一种用于训练物理信息神经网络和运算网络的新型多保真度框架，通过迭代过程中逐步增加模型的表达能力，该方法的迭代特性使我们能够逐步学习直接难以学习的解的特征，通过非线性摆、波动方程和粘性Burgers方程等基准问题，我们展示了如何通过堆叠方法来提高物理信息神经网络和运算网络的准确性和减少所需的规模。

堆叠网络提升物理信息训练：神经网络和深度算子网络的应用