Analyses of the Differentiable-programming Paradigm for Learning Physics-constrained Surrogate Models

Surrogate models of PDEs are an important area of research for applications where rapid, accurate predictions are desired with low computational costs. Deep learning is a popular approach, but they typically lack the strong physical constraints which are intrinsic in PDEs. Furthermore, PDEs such as the Navier-Stokes equations often exhibit chaotic non-local dynamics, which are considerably harder to model than the local dynamics seen in several canonical PDEs. In this work, we present differentiable programming-based strategies as an alternative to learn such dynamics by training neural networks embedded directly inside the PDE structure. In particular, we represent the nonlinear and non-local terms as neural networks and use backpropagation to train them, while simultaneously solving the surrogate PDE in the forward pass. Finally, we investigate the properties of the learned surrogate PDEs, including their sensitivity to system noise, external forcing, impact on prediction accuracy; and comment on potential applications.