Neural Network Trained Schemes for Efficient Simulation of Complex Physics using Adaptive Mesh Refinement

Numerical simulation technology now play an equal role to theory and physical experiments in discovery driven engineering research.

These simulations are usually based on partial differential equations, where the most challenging ones are nonlinear, support boundary layers, discontinuities, wave propagation and often performed in domains with both smooth and complex geometries.

One of the main computational challenges in these problems lies in combining local high resolution requirements (many grid points for accuracy) with low resolution demands (few grid points for efficiency) while retaining stability. These resolution requirements require some type of adaptive mesh refinement (AMR).

In this project we will use deep feedforward artificial neural networks (ANNs) to construct difference operators on summation-by-parts (SBP) form to be used on arbitrary multidimensional meshes with high accuracy.

The SBP form of the operators lead to provably stable schemes to be used in an efficient AMR procedure (that retain the stability).

The new trained SBP operators will have adapted weights for the specific mesh, hence avoiding a possible mismatch between the mesh structure and the approximation technique. Our proposed procedure is based on: 1. Extensive previous experience of constructing schemes on SBP form and 2.

New experiences from VR financed work (2018-05084-VR) where both ANN generated SBP operators on general one-dimensional domains and stable AMR procedures were developed.