Partial differential equations (PDEs) have been studied for centuries and have seen an impressive utilization in scientific computing (SC) during the last sixty years due to increasingly powerful computers. Alongside the utilization, a powerful theoretical foundation has been developed and this foundation ensures both efficient computations and accurate results. In the last ten years, an explosion of usage of machine learning (ML) techniques in the form of deep neural networks (DNNs) has demonstrated a wide range of successes due to both high-performance computing and vast amounts of available data. Despite the similarities between the different areas, the synergies' effects have so far been sparse, in particular on the theoretical level. This proposal aims to bridge the gap between these areas. The project addresses challenges on the long-term horizon in the IKTPluss program. There are three crucial developments in the theory of computational methods for PDEs that should be merged with DNN. The first is the development of more reliable and robust machine learning techniques by exploiting multigrid (MG) techniques developed for the solution of PDEs. The second is the integration of DNNs into an MG framework. And the third topic concerns the integration of DNNs and FEM to enable the learning of computational models. These theoretical developments should be accompanied by software development and relevant applications. Here, applications in biology and medicine are of particular importance because the underlying principles are often not well understood. In particular, we will investigate a novel mechanism related to Alzheimer´s disease.
University of Oslo