Applied Mathematics Seminar


DATE2024-02-23@11:00-12:00

PLACEƾǨt] 1F3174Ы

SPEAKERProfessor Jack Xin]University of California, Irvine, USA^

TITLEDeepparticle: Learning Multiscale Pdes with Data Generated from Interacting Particle Methods

ABSTRACT Multiscale time dependent partial differential equations (PDE) are challenging to compute by traditional mesh based methods especially when their solutions develop large gradients or concentrations at unknown locations. Particle methods, based on microscopic aspects of the PDEs, are mesh free and self-adaptive, yet still expensive when a long time or a resolved computation is necessary.
We present DeepParticle, an integrated deep learning, optimal transport (OT), and interacting particle (IP) approach, to speed up generation and prediction of PDE dynamics through two case studies on transport in fluid flows with chaotic streamlines: 1) large time front speeds of Fisher-Kolmogorov-Petrovsky-Piskunov equation (FKPP); 2) Keller-Segel (KS) chemotaxis system modeling bacteria evolution in the presence of a chemical attractant.
Analysis of FKPP reduces the problem to a computation of principal eigenvalue of an advection-diffusion operator. A normalized Feynman-Kac representation makes possible a genetic IP algorithm to evolve the initial uniform particle distribution to a large time invariant measure from which to extract front speeds. The invariant measure is parameterized by a physical parameter (the Peclet number). We train a light weight deep neural network with local and global skip connections to learn this family of invariant measures. The training data come from IP computation in three dimensions at a few sample Peclet numbers. The training objective being minimized is a discrete Wasserstein distance in OT theory. The trained network predicts a more concentrated invariant measure at a larger Peclet number and also serves as a warm start to accelerate IP computation. The KS is formulated as a McKean-Vlasov equation (macroscopic limit) of a stochastic IP system. The DeepParticle framework extends and learns to generate various finite time bacterial aggregation patterns.
Joint work with Zhongjian Wang (Nanyang Tech Univ, Singapore) and Zhiwen Zhang (University of Hong Kong).