Neural network-based time stepping scheme for multiscale partial differential equations

The macroscopic coherent behavior in complex systems usually may occur from the interactions of microscopic agents, such as molecules, cells and individuals in a population, between themselves and their environment resulting in the multiscale behavior of such systems. Multiscale systems are computationally expensive to simulate as they exhibit behavior at multiple spatial and temporal scales. Capturing the fastest time scale requires taking extremely small steps in numerical time stepping algorithms while capturing the slow scale behavior necessitates long term solution, thus rendering the task computationally prohibitive. In this paper, we demonstrate the application of deep learning techniques to derive an accurate time stepping scheme for multiscale systems. The proposed technique can capture the multiscale PDE behavior with high accuracy while being computationally tractable. We demonstrate the scheme on two benchmark nonlinear PDEs that exhibit multiscale behavior.