Jakob Zech (Universität Heidelberg)

Neural and spectral operator surrogates

In this talk we discuss the use of neural network based operator surrogates to approximate smooth maps between infinite-dimensional Hilbert spaces. Such surrogates have a wide range of applications and can be used in uncertainty quantification and parameter estimation problems in fields such as classical mechanics, fluid mechanics, electrodynamics, earth sciences etc. In this case, the operator input represents the problem configuration and models initial conditions, material properties, forcing terms and/or the domain of a partial differential equation (PDE) describing the underlying physics. The output of the operator is the corresponding PDE solution. We will also present an alternative approach using interpolation, which allows for deterministic construction and eliminates the need for training the network weights. In both cases, algebraic and dimension-independent convergence rates are obtained.

Markus Schmidtchen (TU Dresden)

Interacting Species Across Scales — Analysis and Applications

Travelling pulses of bacteria, pigment cells, whole patches of tissue, flocks of birds, fish schools, or even pedestrians — non-locally interacting agents are ubiquitous in nature. Often, it is possible to observe and model simple behavioural rules between two (indistinguishable) individuals based on biological or social forces. These interactions between any two individuals are typically referred to as first principles and lead to very rich and complex behaviours as soon as many individuals are involved. In the first part of the talk, we propose a non-local model for two interacting species and discuss pattern formation and phase segregation effects with applications to tissue growth and morphogenesis in zebrafish. We proceed by discussing the role of size-exclusion effects and shape on collective dynamics. We conclude the seminar by presenting formal singular limits (only hinting at the analytical intricacies) that establish bridges between different modelling paradigms of multi-agent systems.

Jutta Rogal (FU, NYU)

Exploring free energies with deep generative models

Computing free energy differences is a computationally demanding task, requiring a rigorous sampling of the phase space. Here, we train a machine learning model based on normalizing flows to map between probability distributions of condensed phase systems at different thermodynamic conditions. Using the trained flow model, a large number of uncorrelated configurations can easily be generated to efficiently estimate ensemble averages. This allows us to assess free energy differences over a wide range of temperatures and pressure, needed to evaluate the relative stability of different phases and reconstruct phase diagrams.