Exploiting structural knowledge in (nonlinear) PDE problems for efficient deployment of the learning abilities of ANNs

Artificial neural network (ANN) codes for learning the dynamics of partial differential equations (PDEs) provide impressive results, such as the data-driven discovery of suitable coordinates and of governing (effictive equations. We aim to exploit the learning abilities of ANNs in order to accelerate numerical simulations by using structural knowledge of the PDE. In many time-dependent PDE problems we know for structural reasons (e.g. the presence of symmetries, localizations or oscillations) that the time-consuming part of simulations appears over and over again, but is of very similar nature. Thus, we expect that learning this part of the behavior and replacing it in the simulations by an adaptively used ML „black box“ leads to a significant reduction of simulation time. Since each subsequent simulation in the black box requires less computational costs, it makes this approach suitable for systems on spatially extended domains, for systems exhibiting solutions of low regularity or for system with a multiple sailing character. In this project we plan to apply the approach in such systems and show that other (uninformed) numerical schemes fail as either a very large grid or a very fine grid size are necessary. In addition, from the learned, or discovered, dynamics in the black box and the associated effective equations we hope to derive new analytical insights.