Novel Machine Learning Methods for Resolving Coherent Structures in Multiscale Physical Simulations
Abstract
Recent advances in modern machine learning have uncovered novel ways to simulate and predict high dimensional dynamics. This has led to an effort to apply these models to nonlinear partial differential equations with a range of applications in fluid dynamics, weather prediction, and complex dynamical systems. The goal of many of these research areas is to go to higher spatial resolution or farther in time using prior knowledge of the behavior of the system, rather than using higher resolution or longer time simulation with conventional methods for solving nonlinear PDE. This project will address problems in nonlinear dynamics for which direct numerical simulation (DNS) may not provide sufficient resolution to provide answers to fundamental problems in PDE and in related scientificproblems. We focus on formation and persistence of coherent multiscale structures such as finite or infinite time singularities.Thedynamics of the underlying DNS of a nonlinear PDE, especially in the case of finite time singularities or pattern forming dynamics, necessitate irregular time series observation. For example, in the case of finite time singularities, decreasing step size is needed as one approaches the formation of the coherent structure. Coarsening dynamics in phase separation is well-known to have power-law scaling, requiring adaptive time-stepping methods and resulting in irregular time intervals for changes of state of the solution. We propose to leverage novel heavy-ball neural ODE methods (Neurips 2021), designed for irregularly sampled time series data combined with graphical models for high dimensional data structures as are often required for DNS of grid refinement for nonlinear PDE.We propose some test problems for this research plan that will allow us to compare against recent results from the Fourier Neural Operator method based on pseudospectral direct numerical simulation. These test problems include a modified Kuramoto-Sivashinsky model that produces finite time singularities, a thin film equation that has known results showing that Fourier-based methods fail to capture the inner structure of singularities, and phase-field dynamics which are well-known to work well with spectral methods yet suffer from resolution problems when the diffuse interface parameter vanishes. Computational advances from this research program may have a significant impact on our understanding of complex physical systems with multiple time and spatial scales.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 29, 2023
- Source ID
- N000142312565
Entities
People
- Andrea Bertozzi
Organizations
- Office of Naval Research
- United States Navy
- University of California, Los Angeles