Research

Fluctuations are ubiquitous in real world contexts and in key technological challenges, ranging from thermal fluctuations in physical systems, to algorithmic stochasticity in machine learning, to fluctuations caused by small-scale weather patterns in climate dynamics. At the same time, such complex systems are subject to an abundance of influences, and depend on a large variety of parameters and interactions.

A systematic understanding of the interplay of stochasticity and complex dynamical behavior aims at unveiling universal properties, irrespectively of the many details of the concrete systems at hand.

In our research group we aim to derive and analyse universal concepts in scaling limits of such complex systems, capturing not only their average behavior, but also their fluctuations.

This entails mathematical questions linking the fields of partial differential equations, stochastic analysis, mathematics of machine learning, probability theory and dynamical systems.

Some of the key mathematical challenges addressed in our research are

Mathematical challenges

Nonlinear PDEs

Nonlinear partial differential equations provide the continuum language for systems whose response depends on their current state.

SPDEs

Stochastic partial differential equations describe spatially extended systems under random forcing.

Stochastic Dynamics

Stochastic dynamics asks how random systems evolve over long times and how individual sample paths organize into coherent behaviour.

Numerics

Our numerical programme develops reliable approximations for nonlinear and stochastic systems whose conservation laws, geometry or multiscale structure matter to the observed behaviour.

Main fields of application

Machine Learning

We investigate the mathematical foundations of machine learning through the dynamics, geometry and thermodynamics of learning algorithms.

Non-equilibrium Statistical Mechanics, Interacting Particle Systems and Fluctuating Hydrodynamics

Many macroscopic laws arise from large systems of randomly interacting particles far from equilibrium.

Stochastic Fluid Dynamics

Stochastic fluid dynamics studies the interaction of nonlinear transport, dissipation and random forcing in fluids and interfaces.

Sampling

Sampling turns probability distributions into computable random representatives.

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