Mathematical Developments in GFD,
Idealised Models,
AI-day

In this talk I will discuss recent results in Mathematics+AI. I will describe several concrete instances in which AI systems have contributed to genuine mathematical and scientific results, ranging from the discovery of new objects to the optimization of the numerical kernels that underpin modern computation. A recurring theme is that these tools do not replace mathematical reasoning but extend the range of problems on which it can be brought to bear, often by searching spaces that are too large or too irregular for human intuition alone.

Many low-mass stars like the Sun host periodic, oscillatory magnetic fields that lead to variable levels of stellar activity and variations of space weather, affecting habitability and detection of exoplanets. Due to the intrinsic difficulties of modelling stellar magnetohydrodynamics at all scales, realistic numerical simulations of this process are very challenging and their reduced-order models of oscillatory dynamos are of interest. In this work, we develop a framework to recover such models directly from numerical data using a combination of Hankel Dynamic Mode Decomposition (DMD) to identify magnetic structures, and Sparse Identification of Nonlinear Dynamics (SINDy) to model their dynamics, and compare it to classic mathematical method of weakly nonlinear analysis (WNL). We implement this approach on a one-dimensional idealized mean-field dynamo model parametrizing the main components of convective dynamo in a low-mass star, helical convection and differential rotation.

Reduced-order models offer computationally efficient approximations of complex systems, enabling multi-query tasks in design and optimisation with low cost and sufficient accuracy. Data-driven strategies are particularly appealing when underlying models are inaccessible or too expensive to evaluate, and recent advances in AI-based architectures have naturally entered this space. However, these architectures still face challenges when confronted with systems exhibiting variable dynamics, bifurcations, or chaotic behaviour. In this talk, we present a shift in perspective that unifies complex dynamical systems with nonintrusive, data-driven reduced-order modelling approaches, thereby broadening the range of applications that can be addressed effectively.

John von Neumann is often quoted as saying "with four parameters I can fit an elephant, and with five I can make him wiggle his trunk." The implication seems to be that physical models should contain only a handful of parameters. A century later, however, we seem happy to use physics-agnostic neural networks containing millions of parameters. What would von Neumann say? How should physical modellers respond?
In this talk, I will show that von Neumann's quote is more nuanced than it sounds. I will then frame a response within a Bayesian framework, in which physical principles such as conservation of mass, momentum, and energy are treated as high quality prior information, with quantified uncertainty, expressed as PDEs or low order models. The information content of data can then be quantified and the likelihood of different candidate models can be compared after the data arrives. I will show how Bayesian inference becomes computationally tractable when combined with adjoint methods. I will demonstrate this through assimilation of 3D Flow-MRI data in complex geometry into Finite Element CFD. The main message of the talk is "keep the physics in the model if you can."

In this talk, we'll introduce denoising diffusion models, a class of generative models that rely on additive Gaussian white noise denoising. We'll explain the link with particle-based method of the heat equation in high-dimension. We'll apply these techniques to numerical of multi-layer QG equation in a double-gyre setting, which is an idealized model of the North-Atlantic ocean with a western-boundary (gulf-stream like) current, viewing the numerical solutions of QG equations as a stochastic process that we learn without using explicitly physical priors.

Ocean climate modelling can now resolve geostrophic turbulence rather than merely parametrize it. With this step the ocean has taken the stage as a genuinely turbulent fluid - a remarkable achievement that also marks the end of the "geostrophic era." That ending is equally a beginning, and one that falls in line with the technological shift toward GPU and AI computing. This talk describes new computational approaches to ocean modelling, new experimental strategies and reflects on how machine learning can be integrated into this endeavour.

Techniques of machine learning (ML) find a rapidly increasing range of applications touching upon many aspects of everyday life. They are also used with enthusiasm to close gaps in our scientific knowledge by data-based modeling. I have followed these developments with interest, concern, and mounting disappointment. When these technologies take over decisive functionality in safety-critical applications, we should know how to guarantee their compliance with pre-defined guardrails. Moreover, when they are utilized as building blocks in scientific research, it would violate scientific standards if these building blocks were used without a thorough understanding of their functionality, including inaccuracies, uncertainties, and other pitfalls. In this context, I will juxtapose (a subset of) deep neural network methods with the family of entropy-optimal ML techniques developed recently by Illia Horenko (RPTU Kaiserslautern-Landau) and colleagues.