Predicting parametric spatiotemporal dynamics by multi-resolution PDE structure-preserved deep learning*

Although recent advances in deep learning (DL) have shown a great promise for learning physics exhibiting complex spatiotemporal dynamics, the high training cost, unsatisfied extrapolability, and poor generalizability in out-of-sample regimes significantly limit their applications in science/engineering problems. A more promising way is to leverage physical prior to develop physics-informed deep learning (PiDL) frameworks. In most existing PiDL works, e.g., physics- informed neural networks, the physics is mainly utilized to regularize neural network training by incorporating governing equations into the loss function. In this work, we propose a new direction to leverage physics prior knowledge by baking the mathematical structures of governing equations into the neural network architecture design. In particular, we develop a novel PDE-preserved neural network (PPNN) for rapidly predicting parametric spatiotemporal dynamics, given the governing PDEs are (partially) known. The discretized PDE structures are preserved in PPNN as convolutional residual network blocks, which are formulated in a multi-resolution setting. The effectiveness and merit of the proposed methods have been demonstrated over a handful of spatiotemporal dynamical systems governed by unsteady PDEs.