Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems

TL;DR

This paper introduces a probabilistic deep learning framework for coarse-graining high-dimensional dynamical systems, incorporating physical constraints to improve accuracy, reduce data needs, and enable uncertainty quantification without predefined coarse variables.

Contribution

It develops a novel probabilistic state-space model that integrates physical constraints as virtual observables, eliminating the need for explicit restriction operators and derivative calculations.

Findings

Successfully applied to particle systems demonstrating accurate coarse-graining.

Capable of reconstructing full system evolution from coarse data.

Reduces data requirements by enforcing physical laws.

Abstract

Data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a data-based, probablistic perspective that enables the quantification of predictive uncertainties. One of the outstanding problems has been the introduction of physical constraints in the probabilistic machine learning objectives. The primary utility of such constraints stems from the undisputed physical laws such as conservation of mass, energy etc. that they represent. Furthermore and apart from leading to physically realistic predictions, they can significantly reduce the requisite amount of training data which for high-dimensional, multiscale systems are expensive to obtain (Small Data regime). We formulate the coarse-graining process by employing a probabilistic state-space model and account for the aforementioned equality constraints as virtual observables in the associated densities. We demonstrate how probabilistic inference tools can be employed to identify the coarse-grained variables in combination with deep neural nets and their evolution model without ever needing to define a fine-to-coarse (restriction) projection and without needing time-derivatives of state variables. Furthermore, it is capable of reconstructing the evolution of the full, fine-scale system and therefore the observables of interest need not be selected a priori. We demonstrate the efficacy of the proposed framework by applying it to systems of interacting particles and an image-series of a nonlinear pendulum.