A digital twin is a virtual representation of real-world physical objects. Through accurate and streamlined simulations, it effectively enhances our understanding of the real world, enabling us to predict complex and dynamic phenomena in a fraction of the time. In this talk, we will explore real-world applications of AI-based partial differential equation (PDE) solvers in various fields. Additionally, we will examine how such AI can be utilized to facilitate downstream tasks related to PDEs.

While deep learning has many remarkable success stories, finding a satisfactory mathematical explanation on why it is so effective is still considered an open challenge. One recent promising direction for this challenge is to analyse the mathematical properties of neural networks in the limit where the widths of hidden layers of the networks go to infinity. Researchers were able to prove highly-nontrivial properties of such infinitely-wide neural networks, such as the gradient-based training achieving the zero training error (so that it finds a global optimum), and the typical random initialisation of those infinitely-wide networks making them so called Gaussian processes, which are well-studied random objects in machine learning, statistics, and probability theory. These theoretical findings also led to new algorithms based on so-called kernels, which sometimes outperform existing kernel-based algorithms. The purpose of this talk is to explain these recent theoretical results on infinitely wide neural networks. If time permits, I will briefly describe my work in this domain, which aims at developing a new neural-network architecture that has multiple nice theoretical properties in the infinite-width limit. This work is jointly pursued with Fadhel Ayed, Francois Caron, Paul Jung, Hoil Lee, and Juho Lee.

Tree decompositions are a powerful tool in both structural graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph. Tree decompositions have traditionally been used in the context of forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.

Collective cell movement is critical to the emergent properties of many multicellular systems including microbial self-organization in biofilms, wound healing, and cancer metastasis. However, even the best-studied systems lack a complete picture of how diverse physical and chemical cues act upon individual cells to ensure coordinated multicellular behavior. Myxococcus xanthus is a model bacteria famous for its coordinated multicellular behavior resulting in dynamic patterns formation. For example, when starving millions of cells coordinate their movement to organize into fruiting bodies – aggregates containing tens of thousands of bacteria. Relating these complex self-organization patterns to the behavior of individual cells is a complex-reverse engineering problem that cannot be solved solely by experimental research. In collaboration with experimental colleagues, we use a combination of quantitative microscopy, image processing, agent-based modeling, and kinetic theory PDEs to uncover the mechanisms of emergent collective behaviors.

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Professor of Bioengineering & BioSciences, Associate Chair of Bioengineering, Rice U