Time-integrating Physics-informed Neural Networks (TPINNs) for Identifying and Modelling Time-dependent PDEs

Partial Differential Equations (PDEs) are important tools in science and engineering for describing the dynamics in systems of interest. Some PDEs can be derived from first principles, but their coefficients may be unknown due to the obscure complexity of the systems. Physics-informed Neural Networks (PINNs) have been widely used for identifying coefficients and predicting dynamics of PDEs from data describing the dynamics. In this paper, we propose a novel type of PINNs for time-dependent PDEs, namely, Time-integrating PINNs (TPINNs), which define the loss function as a tailor-designed time-integration functional. In TPINNs, various kinds of time-integrator can be incorporated into the loss function, for example, variational time-integrator and operator splitting schemes. We performed detailed experiments on two famous time-dependent PDE problems, which include the wave equation, and a relatively more complicated PDE, the incompressible Euler equations. Our extensive empirical evidence reveals that TPINNs outperform PINNs by achieving substantially 1) higher accuracy and stability in coefficient estimation and 2) more accurate and stable dynamics prediction, 3) both without any additional cost on the training data size and parameter space size and at comparably similar level of training time. It is strongly believed TPINNs have unlimited potential in coefficient estimation and dynamics prediction for time-dependent PDEs in a wide range of complicated applications.