At this colloquium, we are happy to welcome:
Mark Tuckerman (New York University)
Crystal Math: Rapid and accurate prediction of molecular crystal structures and properties using topological and simple physical descriptors, informatics, and machine learning strategies
The different solid structures or polymorphs of atomic and molecular crystals often possess different physical and chemical properties. Structural differences between organic molecular crystal polymorphs can affect, for example, bioavailability of active pharmaceutical formulations, the lethality of contact insecticides, and diffusive behavior in host-guest systems. Such differences can also influence the behavior of smart materials, such as self-healing crystals or propulsion mechanisms in thermomechanical or so-called “jumping” crystals. In metallic crystals, structural differences may determine how different phases may be used in electronic device applications. Crystallization conditions can influence polymorph selection, making an experimentally driven hunt for polymorphs difficult. These efforts are further complicated when polymorphs initially obtained under a particular experimental protocol “disappear” in favor of another polymorph in subsequent repetitions of the experiment. Theory and computation can potentially play a vital role in mapping the landscape of crystal polymorphism. Traditional methods for predicting crystal structures and investigating solid-solid phase transformation behavior face their own challenges, and therefore, new approaches are needed. In this talk, I will show, by leveraging concepts from mathematics, specifically geometry and topology, in combination with simple physical principles, database processing, and machine learning strategies, including autoencoders and deep graph neural networks, that we have been able to develop a new framework, which we are calling “Crystal Math” that represents a new paradigms in our ability to predict molecular crystal structures and crystal properties, orders of magnitude faster and with far fewer resources than more traditional methods.
Illia Horenko (Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau)
An entropy-optimal path to humble AI
Progress of AI has led to a creation of very successful, but by no means humble models and tools, especially regarding: (i) the huge and further exploding costs and resources they demand, and (ii) the over-confidence of these tools with the answers they provide.
A novel mathematical framework for a non-equilibrium entropy-optimizing reformulation of the Ising model and Boltzmann machines based on the exact law of total probability will be introduced in this talk – hinting at the weak spots of the state-of-the-art tools [1].
Proposed Entropy Optimal Network paradigm (EON) results in the highly-performant, but orders-of-magnitude cheaper, gradient-descent-free AI learning instruments, with mathematically-justified existence and uniqueness criteria - and equipped with the answer confidence/reliability measures.
Comparisons to the state-of-the-art AI tools (including the most modern foundational transformer models like TabPFN) - on a set of synthetic mathematical problems with varying complexity, reveal that the proposed EON method allows more performant and slim models. The descriptor lengths of the obtained EON models are shown to be asymptotically close to the intrinsic Kolmogorov complexity bounds for the underlying mathematical problems.
On a practical side, it will be demonstrated that applying EON to historical climate data results in models with systematically higher climate prediction skills for the onsets of La Nina and El Nino climate phenomena, requiring just around 10 years of climate data for training - a small fraction of what is necessary for contemporary climate prediction tools.
Finally, it will be demonstrated that the computational performance of entropy-minimizing AI tools can further be busted by several orders of magnitude - deploying very close, compact and computationally very cheap rational approximations of entropic measures [2].
Literature:
[1] D. Bassetti, L. Pospíšil, M. Groom, T. O’Kane, and I. Horenko, “An entropy-optimal path to humble AI”, arXiv:2506.17940 [cs.LG] , 2025
[2] I. Horenko, D. Bassetti, L. Pospíšil, “Fast, close and non-singular approximations of entropic measures”, DOI:10.48550/arXiv.2505.14234, 2025
André Schlichting (Universität Ulm)
Singular Limit Analysis of Training with Noise Injection
Many training algorithms inject some form of noise during the training process. A classic example is mini-batch noise in stochastic gradient descent, but other examples include dropout, data augmentation, "noise nodes," "label noise," and input-data noise.
The additional noise is believed to improve generalization performance. However, there is little mathematical understanding of how this is achieved. In this talk, I will present a recent work (arXiv:2404.12293) with Mark Peletier and Anna Shalova (TU/e) in which we analyze a fairly general class of iterative training schemes with noise injection. In the limit of small noise, we prove that the training process, appropriately rescaled in time, converges to solutions of an auxiliary evolution equation. The limit equation is a gradient flow driven by a functional for which we obtain an explicit expression, thus opening the door to understanding the different types of regularization generated by different types of noise injection.