Research projects

Ongoing

ERC

Time-Evolving Stochastic Manifolds

Uncertainty is all around us and caused, for example, by the nature of a problem as in quantum mechanics, the lack of our precise knowledge as in porous media, or inaccuracies in measurements as in experiments with imperfect equipment. While traditionally and due to the lack of computing power, science and technology relied on deterministic models, recent developments allow to include randomness. This trend requires efficient simulation methods for models with uncertainty. In space-time problems such as moving biological cells and the surface of the ocean, the randomness could be modeled by a stochastic process given explicitly or described implicitly by stochastic partial differential equations. Fast and accurate methods for sampling the stochastic processes are the key when computing statistical quantities of the advanced models.

The main contribution of the ERC-funded StochMan project is the development of a theoretical framework for evolving stochastic manifolds and their efficient simulation with analyzed algorithms. Special emphasis is paid to the situation when the evolving stochastic manifold is a moving surface disturbed by external forces and described by stochastic partial differential equations.

Team: David Cohen, Erik Jansson, Annika Lang, Stig Larsson, Ioanna Motschan-Armen, Björn Müller, Andrea Papini, Mike Pereira

Outreach: Announcement@Chalmers (January 31, 2023), ECMI blog (September 2023)

VR

Efficient Approximation Methods for Random Fields on Manifolds

Uncertainty is all around us and caused, for example, by the nature of a problem as in quantum mechanics, the lack of our precise knowledge as in porous media, or inaccuracies in measurements. While traditionally and due to the lack of computing power, science and technology relied on deterministic models, new developments allow to include randomness in the models. The new trend requires efficient simulation methods to sample the randomness. In spatial problems such as weather predictions, ground water flows, and material surfaces, the randomness should be modeled by a random field. The speed and the quality of the used sampling methods determine if the noisy models are applicable outside of academia.The purpose of this project is to develop and analyze efficient algorithms that approximate random fields with a given precision based on a prescribed covariance. By the end of the project new algorithms for nonstationary and anisotropic random fields on manifolds are delivered. We will start with Gaussian fields and consider generalizations at a later stage of the project. Based on earlier results for isotropic random fields on spheres, we are in a good starting position to extensions to Riemannian manifolds such as surfaces.The project will finance the project leaders research time and a PhD student. Local and international collaborations complete the team and gather all necessary competences that guarantee the success of the project.

Team: Erik Jansson, Annika Lang, Ioanna Motschan-Armen, Mike Pereira

CHAIR

Stochastic Continuous-Depth Neural Networks

We advance the understanding of deep neural networks through the investigation of stochastic continuous-depth neural networks. These can be thought of as deep neural networks (DNN) composed of infinitely many stochastic layers, where each single layer only brings about a gradual change to the output of the preceding layers. We will analyse such stochastic continuous-depth neural networks using tools from stochastic calculus and Bayesian statistics. From that, we will derive practically relevant and novel training algorithms for stochastic DNNs with the aim to capture the uncertainty associated with the predictions of the network. This project is supported by Chalmers AI Research Centre (CHAIR) including Oskar Eklund's PhD position.

Team: Oskar Eklund, Annika Lang, Moritz Schauer