SciML - Scientific Computing & Machine Learning

Artificial intelligence in the form of self-learning neural networks has in the recent years demonstrated extreme power and success in a series of applications such as image and sound recognition and classification as well as super-human powers in games such as chess. Nevertheless, even when a network is apparently properly trained, the networks make odd mistakes that humans would not do. Therefore, the application of such networks in mission-critical situations such as for instance, medicine and military applications or even self-driving is questionable. In contrast, within modeling based sciences such as e.g. physics, engineering and chemistry, there has been established a robust and accurate framework of analysis that often guaranties safe application with high precision. In particular, the mathematical framework developed for the analysis of partial differential equations provides a foundation for error analysis and quantification which when combined with high-performance computing enable realistic simulations of a wide-range of phenomena. In this project, we will explore ways of combining and extending traditional methods in scientific computing with neural networks in order to extend the flexibility of the traditional methods as well as increasing the robustness of today?s learning techniques. The primary focus so far has been to unravel instabilities in the couplings between neural nets and finite elements. We have combined FEM and NN in a variety of ways and unraveled instabilities and challenges in complicated settings. That said, the framework appears relatively robust in applications. Physics-informed neural networks are promising for applications within nevroscience. Vi have recently succeeded to establish some theoretical considerations regarding the foundations of neural nets that we now hope to publish. We have succeded with elaborate modeling of the so-called glymphatic system, a well-known system in neuroscience, responsible for clearance during sleep and have received attention from the mass media.

The project has enabled research on modeling and learning based method for applications that has led to a substantial number of scientific papers in top journals, along with one book and associated software tools. Furthermore, the project has enabled us to derive methods and corresponding analysis for physics informed networks, sharper than ever before, along with tools for practical machine learning within image based neuroscience. Some of the results have even been picked up by public media in Norway, such as VG and NRK.

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 with 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) have 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 a MG framework. And the third topic concerns the integration of DNNs and FEM to enable learning of computational models. These theoretical developments should be accompanied with 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 in which the PI is on the forefront of current research.