  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.
