A coupled space-time framework aided by physics-informed neural networks to accelerate fluid flow simulations*

The idea of time parallelism allows an alternate strategy to accelerate numerical simulations when it is no longer feasible to leverage the benefits of space parallelism. The strategy used in time-parallel methods essentially involves using a coarse-grid propagator/predictor method to estimate a good initial guess of the solution across the time horizon. The time domain can then be decomposed and run in parallel using the initial guess obtained using the coarse grid propagator. This approach has been discussed in several parallel-in-time (PinT) methods such as the Parareal algorithm, where a numerical solver with a very coarse time-step is used as the predictor method.

In this work, we discuss time parallelism with another class of methods called the integrated/coupled space-time framework or simply space-time framework, where time is treated as a physical dimension and the original initial value problem becomes a boundary value problem. We extend the idea of obtaining a good initial guess (using a predictor method) to the space-time framework. The use of a predictor method to accelerate simulations in the space-time framework has not been discussed in the literature. Instead of using standard numerical solvers, we propose the use of Physics-Informed Neural Networks (PINN) to generate the initial trajectory of the space-time solution. Once trained, neural networks can generate a wide variety of initial conditions for a minimial computational cost. We demonstrate the use of a PINN framework as a predictor method to accelerate numerical simulations by experiments on two model problems in one and two spatial dimensions: (a) diffusion, and (b) Burgers' equation. We show that the use of a PINN as a predictor helps the numerical solution converge significantly faster and the performance improves as the space-time mesh is refined.