This Week in Mathematics


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Entries for this week: 3

Wednesday January 22, 2025

Biomathematics Journal Seminar
Lotka-Volterra Predator-Prey Model With Periodically Varying Carrying Capacity
- Bhargav Karamched, FSU
Time: 5:00 Room: Dirac Library

Mathematics Colloquium [url]
An Adversarial Deep Learning approach using Natural Gradients for solving Partial Differential Equations
- Shu Liu, UCLA
Time: 3:05 Room: 101
Abstract/Desc: We propose a scalable, preconditioned primal-dual algorithm for solving partial differential equations (PDEs). By multiplying the equation with a test function, we reformulate it as an inf-sup problem, resulting in a loss function involving lower-order differential operators. To address this saddle point problem, we employ the Primal-Dual Hybrid Gradient (PDHG) algorithm. By introducing suitable preconditioning operators to the metric terms in PDHG proximal steps, we obtain an alternative natural gradient ascent-descent optimization scheme for updating primal and adversarial neural network parameters. These natural gradients are efficiently computed using the Krylov subspace iteration. An a posteriori convergence analysis is established for the time-continuous version of the proposed method. The algorithm is tested on various types of linear and nonlinear PDEs, scaling seamlessly to 50 dimensions. Numerical experiments highlight the method's improved accuracy, efficiency, and stability in convergence when compared to conventional deep-PDE solvers.

Thursday January 23, 2025

Financial Mathematics Seminar
Kakutani’s walk on spheres and efficient methods for solving Laplace and Poisson equation in domains with complex geometry
- Arash Fahim ,
Time: 3:05 Room: LOV 0231
Abstract/Desc: Kakutani’s walk on spheres (WoS) has proven to improve the efficiency of Feynman-Kac formula in solving linear elliptic equations with constant coefficients in bounded domains where the geometry is challenging. Such problems have applications in rendering in computer graphics, simulations in nuclear reactors, and evaluation of capacitance in semi-conductors. Motivated by the recent advancement in solving partial differential equations via deep learning, we introduce a new method, WoS-NN, by integrating WoS into a deep learning framework. Our study shows accurate field estimations, reducing 76.32% errors while using only 8% of path samples compared to the conventional WoS method via Feynman-Kac formula, which saves abundant computational time and resource consumption.

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Department of Mathematics

This Week in Mathematics