Wuchen Li : Mean-Field Games for Scalable Computation and Diverse Applications
- Applied Math and Analysis ( 205 Views )Mean field games (MFGs) study strategic decision-making in large populations where individual players interact via specific mean-field quantities. They have recently gained enormous popularity as powerful research tools with vast applications. For example, the Nash equilibrium of MFGs forms a pair of PDEs, which connects and extends variational optimal transport problems. This talk will present recent progress in this direction, focusing on computational MFG and engineering applications in robotics path planning, pandemics control, and Bayesian/AI sampling algorithms. This is based on joint work with the MURI team led by Stanley Osher (UCLA).
Jacob Bedrossian : Positive Lyapunov exponents for 2d Galerkin-Navier-Stokes with stochastic forcing
- Applied Math and Analysis ( 387 Views )In this talk we discuss our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. The method we proposed combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the stationary measure of the Markov process tracking tangent directions (the so-called "projective process"); and (B) an L1-based hypoelliptic regularity estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving that the projective process satisfies Hörmander's condition. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry.
Jun Kitagawa : A convergent Newton algorithm for semi-discrete optimal transport
- Applied Math and Analysis ( 229 Views )The optimal transport (Monge-Kantorovich) problem is a variational problem involving transportation of mass subject to minimizing some kind of energy, and it arises in connection with many parts of math, both pure and applied. In this talk, I will discuss a numerical algorithm to approximate solutions in the semi-discrete case. We propose a damped Newton algorithm which exploits the structure of the associated dual problem, and using geometric implications of the regularity theory of Monge-Amp{\`e}re equations, we are able to rigorously prove global linear convergence and local superlinear convergence of the algorithm. This talk is based on joint work with Quentin M{\’e}rigot and Boris Thibert.
Linfeng Zhang : Neural network models and concurrent learning schemes for multi-scale molecular modelling
- Applied Math and Analysis ( 220 Views )We will discuss two issues in the context of applying deep learning methods to multi-scale molecular modelling: 1) how to construct symmetry-preserving neural network models for scalar and tensorial quantities; 2) how to efficiently explore the relevant configuration space and generate a minimal set of training data. We show that by properly addressing these two issues, one can systematically develop deep learning-based models for electronic properties and interatomic and coarse-grained potentials, which greatly boost the ability of ab-initio molecular dynamics; one can also develop enhanced sampling techniques that are capable of using tens or even hundreds of collective variables to drive phase transition and accelerate structure search