PHYSICS-AWARE, PROBABILISTIC MODEL ORDER RE-DUCTION WITH GUARANTEED STABILITY

Abstract

Given (small amounts of) time-series' data from a high-dimensional, fine-grained, multiscale dynamical system, we propose a generative framework for learning an effective, lower-dimensional, coarse-grained dynamical model that is predictive of the fine-grained system's long-term evolution but also of its behavior under different initial conditions. We target fine-grained models as they arise in physical applications (e.g. molecular dynamics, agent-based models), the dynamics of which are strongly non-stationary but their transition to equilibrium is governed by unknown slow processes which are largely inaccessible by brute-force simulations. Approaches based on domain knowledge heavily rely on physical insight in identifying temporally slow features and fail to enforce the long-term stability of the learned dynamics. On the other hand, purely statistical frameworks lack interpretability and rely on large amounts of expensive simulation data (long and multiple trajectories) as they cannot infuse domain knowledge. The generative framework proposed achieves the aforementioned desiderata by employing a flexible prior on the complex plane for the latent, slow processes, and an intermediate layer of physics-motivated latent variables that reduces reliance on data and imbues inductive bias. In contrast to existing schemes, it does not require the a priori definition of projection operators or encoders and addresses simultaneously the tasks of dimensionality reduction and model estimation. We demonstrate its efficacy and accuracy in multiscale physical systems of particle dynamics where probabilistic, long-term predictions of phenomena not contained in the training data are produced.

1. INTRODUCTION

High-dimensional, nonlinear systems are ubiquitous in engineering and computational physics. Their nature is in general multi-scalefoot_0 . E.g. in materials, defects and cracks occur on scales of millimeters to centimeters whereas the atomic processes responsible for such defects take place at much finer scales (Belytschko & Song, 2010) . Local oscillations due to bonded interactions of atoms (Smit, 1996 ) take place at time scales of femtoseconds (10 -15 s), whereas protein folding processes which can be relevant for e.g. drug discovery happen at time scales larger than milliseconds (10 -3 s). In Fluid Mechanics, turbulence phenomena are characterized by fine-scale spatiotemporal fluctuations which affect the coarse-scale response (Laizet & Vassilicos, 2009) . In all of these cases, macroscopic observables are the result of microscopic phenomena and a better understanding of the interactions between the different scales would be highly beneficial for predicting the system's evolution (Givon et al., 2004) . The identification of the different scales, their dynamics and connections however is a non-trivial task and is challenging from the perspective of statistical as well as physical modeling.



With the term multiscale we refer to systems whose behavior arises from the synergy of two or more processes occurring at different (spatio)temporal scales. Very often these processes involve different physical descriptions and models (i.e. they are also multi-physics). We refer to the description/model at the finer scale as fine-grained and to the description/model at the coarser scale as coarse-grained.

