Scientific Computing is a field concerning fast and reliable simulations for various applications. This involve solving complicated problems including multi-scale physics, inverse problems, data assimilation, and optimal control. It is highly interdisciplinary and requires tools from pure mathematics, numerical analysis and high-performance computing.
I am particularly interested in combining conventional approaches from numerical analysis with newer methods from machine learning. The final goal is to derive robust, reliable, interpretable, and computationally fast algorithms.
Physics-Informed Machine Learning
Physics-Informed machine learning is a relative new field. It is often referred to as physics-based or domain-aware machine learning. The general idea is to incorporate physical information, often in shape of well-proven models and/or restrictions, in the training of machine learning models. This brings benefits by speeding up the training, decreases the need for very large amount of data, and brings robustness to the final model by ensuring that unphysical events are unlikely to be predicted.
Reduced Order Modeling
Reduced Order Modeling is a field concerning reduction of the computational complexity of numerical computations. The process is often divided into two phase: i) An offline phase, in which a reduced model, often denoted a surrogate model, is computed, and ii) an online phase in which the surrogate model is put to use. A typical online phase would be a multi-query problem, such as uncertainty quantification or inverse problem, or a real-time control problem where very fast solving is of essence.
Much of my research is especially focused on using deep learning in this area. Either for dimensionality reduction or as a surrogate model. Neural networks show a lot of potential in this area due to their ability of dealing with high-dimensional data exceptionally well, their ability to recognise patterns, and their ability to express especially nonlinear maps.