Machine-Learning-based String Method for Minimum Energy Path in Probability Measure Space
Project: Research
Description
This proposal studies the noise-induced transition path between multiple equilibrium distributions in stochastic interacting particle systems. These systems of interacting particles appear in various applications in statistical physics, material science) social science, economics) and machine learning. As the number of particles tends to infinity, the particle distribution has a mean-field limit, described by the Wasserstein gradient flow of well-defined free energy defined on space of probability measures. However, the interactions usually generate multiple stationary measures minimizing the free energy. Then for a finite number of particles, the empirical distribution can switch between these different metastable states in probability space. Such abrupt changes or transitions, usually called rare events, although occurring at a very low rate, have significant impacts at the systemic level. A common practice to understand how transitions happen is to numerically examine the optimal transition path based on the large deviation theory and least action principle. This path can be well described geometrically on the free energy landscape and is called the minimum energy path(MEP). Perhaps the most prominent approach to studying this path is the string method, a gradient-based geometric method to compute the MEP.This project aims to attack two challenges in the string method in the space of probability measures: (1) how to numerically approximate the curve in the infinitedimensional space of probability measures (2) how to devise the computation-efficient dynamics to evolve the path on Wasserstein manifold in the lack of Euclidean metric. We use the recent emergent techniques from scientific machine learning to provide comprehensive and efficient numerical solutions. We propose two methodologies: (1) We take the Eulerian point of view to treat the curve as a static model, and construct the deep neural network as a function of (s, x) jointly for the curve of probability density functions and solve the core problem of training algorithms inspired by the idea from physics-informed neural network(PINN) method; (2) We take the Lagrangian dynamic point of view to explore a new parametric curve of probability measure by assocaiting the curve with a flow map, and discretize the map by normalizing flow technique and devise the string dynamics to find the optimal parameters.These results shall advance the path-finding methods for various free energy on probability space of high dimensional states and enable data-driven investigation of rare events in applications modeled as interactive particle systems.Detail(s)
Project number | 9043582 |
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Grant type | GRF |
Status | Active |
Effective start/end date | 1/01/24 → … |