Encoding physics to learn reaction–diffusion processes

Modelling complex spatiotemporal dynamical systems, such as reaction–diffusion processes, which can be found in many fundamental dynamical effects in various disciplines, has largely relied on finding the underlying partial differential equations (PDEs). However, predicting the evolution of these systems remains a challenging task for many cases owing to insufficient prior knowledge and a lack of explicit PDE formulation for describing the nonlinear process of the system variables. With recent data-driven approaches, it is possible to learn from measurement data while adding prior physics knowledge. However, existing physics-informed machine learning paradigms impose physics laws through soft penalty constraints, and the solution quality largely depends on a trial-and-error proper setting of hyperparameters. Here we propose a deep learning framework that forcibly encodes a given physics structure in a recurrent convolutional neural network to facilitate learning of the spatiotemporal dynamics in sparse data regimes. We show with extensive numerical experiments how the proposed approach can be applied to a variety of problems regarding reaction–diffusion processes and other PDE systems, including forward and inverse analysis, data-driven modelling and discovery of PDEs. We find that our physics-encoding machine learning approach shows high accuracy, robustness, interpretability and generalizability. Reaction–diffusion processes, which can be found in many fundamental spatiotemporal dynamical phenomena in chemistry, biology, geology, physics and ecology, can be modelled by partial differential equations (PDEs). Physics-informed deep learning approaches can accelerate the discovery of PDEs and Rao et al. improve interpretability and generalizability by strong encoding of the underlying physics structure in the neural network.