4:00pm ̶ 4:30pm: Coffee, Drinks & Refreshments
4:30pm ̶ 4:50pm: Professor Jinchao Xu (King Abdullah University of Science and Technology; Director of KAUST-SRIBD Joint Lab for Scientific Computing and Machine Learning; Director of KAUST Innovation Hub in Shenzhen)
Title: Scientific Computing and Machine Learning
Abstract: TBA
4:55pm – 5:15pm: Dr Dan Hill (Hooke Fellow, Mathematical Institute, University of Oxford)
Title: Going Around in Circles: Approximating Fully Localised 2D Patterns via Radial Spatial Dynamics
Abstract: The existence of localised two-dimensional patterns has been observed and studied in numerous experiments and simulations: ranging from optical solitons, to patches of desert vegetation, to fluid convection. And yet, our mathematical understanding of these emerging structures remains extremely limited beyond one-dimensional examples. A key concept in mathematically studying localised patterns is ‘spatial dynamics’, where one treats an unbounded spatial direction as a ‘time-like’ variable and uses tools from dynamical systems to prove the existence of exponentially decaying (in ‘time’) solutions. However, a key question emerges when considering higher spatial dimensions; how can one use such an approach to study patterns that are localised in multiple directions? In this talk, we will focus on 2D patterns that are localised in the radial direction and explore the theory of radial spatial dynamics. We will review prior studies of axisymmetric patterns, before presenting recent results for approximating fully localised patterns with dihedral symmetries. Throughout we will highlight connections to the 1D problem, how new classes of solutions can emerge in this framework, as well as some of the difficulties encountered when considering nonlinear PDEs in polar coordinates.
5:20pm ̶ 5:45pm: Coffee, Drinks & Refreshments
5:45pm ̶ 6:05pm: Dr Daniel Csillag (FGV EMAp, Rio de Janeiro, Brazil)
Title: Random Gradient-Free Optimization in Infinite Dimensional Spaces
Abstract: Functional optimization problems arise in a variety of applications, including statistical learning, optimization-based PDE solvers, and general inverse problems. Though often solved through finite-dimensional gradient descent over a parametrization of the functions, such as neural networks, an interesting alternative is to instead perform gradient descent directly in the function space by leveraging its Hilbert space structure, thus enabling provable guarantees and fast convergence. However, infinite-dimensional gradients are often hard to compute in practice, hindering the applicability of such methods. To overcome this limitation, we propose a new random ‘gradient-free’ method that requires only the computation of directional derivatives and a pre-basis for the Hilbert space domain, i.e., a linearly-independent set whose span is dense in the Hilbert space. This fully resolves the tractability issue, as pre-bases are much more easily obtained than full orthonormal bases or reproducing kernels — which may not even exist — and individual directional derivatives can be easily computed using automatic differentiation. We showcase the use of our method to solve partial differential equations à la physics-informed neural networks (PINNs), where it effectively enables provable convergence.
6:10pm ̶ 6:30pm: Shuchen Guo (Mathematical Institute, University of Oxford)
Title: From Interacting Particles to Partial Differential Equations
Abstract: Interacting particle systems model the dynamics of many agents following simple rules. When the number of particles is large, complex collective behaviour can emerge, often described by continuum quantities such as density and average velocity that satisfy partial differential equations (PDEs). This talk explains how to connect microscopic dynamics to macroscopic equations by taking rigorous scaling limits in which particle models give rise to PDEs, with examples motivated by mathematical physics, mathematical biology, and fluid mechanics.
6:35pm ̶ 7:00pm: Discussion
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