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TILOS Seminar: Kinetic Theory Perspective of Foundation Models for Physics
Speaker: Maarten de Hoop, Rice University
Abstract: We present a kinetic theory perspective of foundation models for physics. We begin with providing a mathematical framework for analyzing transformers. To uniformly address their expressivity, we consider the case that the mappings are conditioned on a context represented by a probability distribution of tokens. That is, transformers become mappings between probability measures. The relevant notion of smoothness then corresponds to continuity in terms of the Wasserstein distance between such contexts. We demonstrate that deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains. We then characterize the conditions on mappings between measures that enable these to be represented in terms of in-context mappings as transformers. The solution map of the Vlasov equation, which is of nonlocal transport type, for interacting particle systems in the mean-field regime for the Cauchy problem satisfies the conditions; conversely, we prove that the measure-theoretic self-attention has the properties that ensure that the infinite depth, mean-field transformer can be identified with a Vlasov flow. Extending this framework from interactions to collisions leads to a further development of structured architectures inspired by Lattice Boltzmann Models, while flow motivates a design based on self-warping.
Bio: Professor Maarten V. de Hoop, Simons Chair in Computational and Applied Mathematics and Earth Science at Rice University, is internationally recognized for his contributions to the mathematical foundations of seismology, wave propagation, and inverse problems. His research bridges microlocal and harmonic analysis, scattering theory, and structured numerical methods with applications to seismic imaging, geophysical inversion, and large-scale computational modeling of acoustic, elastic, and electromagnetic phenomena. De Hoop has been a pioneer in developing techniques to extract subtle information from massive, complex seismic datasets, advancing our ability to probe the Earth’s interior with unprecedented resolution, and more recently has integrated deep learning and data-driven discovery with rigorous mathematical frameworks to open new frontiers in the analysis of multiscale wave phenomena and inverse spectral problems. He is the recipient of the J. Clarence Karcher Award from the Society of Exploration Geophysicists and the Young Scientists Award from the International Society for Analysis, its Applications and Computation, has been elected a Fellow of the Institute of Physics and an External Member of the Finnish Academy of Science and Letters, and has served as associate editor for Inverse Problems, Inverse Problems and Imaging, and the International Journal on Geomathematics.