Abstract
Many living and complex systems exhibit second-order emergent dynamics. Limited experimental access to the configurational degrees of freedom results in data that appear to be generated by a non-Markovian process. This limitation poses a challenge in the quantitative reconstruction of the model from experimental data, even in the simple case of equilibrium Langevin dynamics of Hamiltonian systems. We develop a novel Bayesian inference approach to learn the parameters of such stochastic effective models from discrete finite-length trajectories. We first discuss the failure of naive inference approaches based on the estimation of derivatives through finite differences, regardless of the time resolution and the length of the sampled trajectories. We then derive, adopting higher-order discretization schemes, maximum-likelihood estimators for the model parameters that provide excellent results even with moderately long trajectories. We apply our method to second-order models of collective motion and show that our results also hold in the presence of interactions.
- Received 23 December 2019
- Revised 18 April 2020
- Accepted 22 May 2020
DOI:https://doi.org/10.1103/PhysRevX.10.031018
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Learning from empirical observations is a long-standing goal in physics, but even today it is often harder than expected. Connecting theoretical models to the real observable world is an ongoing challenge, independent of the investigated process. In this spirit, we present a new methodology to correctly extract the equations of motion of small motile objects from available data.
Naive methods fail to learn the equations of motion of these objects since the techniques do not properly take into account the interplay between the features of the continuous-time models (their stochasticity and inertia) and the nature of collected data, which come in the form of time-lapse recorded positions of the moving object. We show this stumbling block in the paradigmatic example of the stochastic harmonic oscillator, and we check that our new proposed systematic approach is able to resolve these issues. Then, we extend our formalism from single-particle problems to systems with many interacting particles, where inferring how particles interact with each other is another source of difficulty.
Our methodology effectively applies to a wide class of dynamical phenomena that can be observed in living systems, like single cells, entire colonies, or even large animal groups (including, in particular, bird flocks and insect swarms), for which an increasing quantity of high-resolution data are now available.
