Uncertainty Quantification by means of Neural Networks
PhD Candidate: Simon Schneider
Supervisors: M. Lukáčová (Mathematics), M. Wand (Computer Science), P. Spichtinger (Atmospheric Physics)
The goal of the project is to investigate the potential of machine learning approaches in the context of fluid dynamics simulations, including uncertain data. The computational costs of such simulations using conventional methods would be otherwise prohibitive. Simon started in October 2020 by studying new solution concepts, their effective numerical approximation, and visualization. First results on a special numerical scheme for the Euler equations combining the idea of viscosity solutions of the Euler equations (limits of solution to the Navier-Sotkes system with vanishing viscosity) with measure valued solutions were published in 2021 (arXiv:pdf/2102.07876). Specifically, Simon carried out the numerical experiments and implemented the required functionality for constructing the proposed approximate solution from sequences of numerical simulations. Afterwards, he tried different machine learning approaches to successfully solve partial differential equations (PDEs). The novel approach was to do this using Physics Informed Neural Networks (PINN), already applied in fluid dynamics. While most machine learning methods rely on an extensive amount of training data, PINNs are promising for problems where the physical laws are (at least partially) known in the form of PDEs. PINNs use these PDEs as constraints to reduce the amount of data needed to train a deep neural network. As simulation data in fluid dynamics is computationally expensive, PINNs offer a better and easier to implement approach than other data-based alternatives. Since October 2021, he started a new framework for proving convergence developed jointly with Eduard Feireisl for numerical solutions of the random Navier-Stokes system.
You can download the original proposal here.