Encoding physics to learn reaction–diffusion processes

Qi Wang

Abstract:

Modelling complex spatiotemporal dynamical systems, such as reaction–
difusion processes, which can be found in many fundamental dynamical
efects in various disciplines, has largely relied on fnding the underlying
partial diferential equations (PDEs). However, predicting the evolution
of these systems remains a challenging task for many cases owing to
insufcient 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–difusion processes and other PDE systems,
including forward and inverse analysis, data-driven modelling and discovery
of PDEs. We fnd that our physics-encoding machine learning approach
shows high accuracy, robustness, interpretability and generalizability