Extracting Signal out of Chaos: Advancements on MAGI for Bayesian Analysis of Dynamical Systems

TL;DR

This paper introduces two new Bayesian methods, pMAGI and PMSP, that improve parameter inference, stability classification, and future trajectory prediction for ODE-based dynamical systems under noisy, sparse data conditions, outperforming existing approaches.

Contribution

The paper presents pMAGI and PMSP, novel Bayesian methods enhancing stability classification and multi-step prediction for dynamical systems, with improved stability, accuracy, and computational efficiency.

Findings

pMAGI improves numerical stability and inference accuracy.

MAGI-based methods can classify system stability probabilistically.

PMSP accurately predicts future trajectories, outperforming PINN-based methods.

Abstract

This work builds off the manifold-constrained Gaussian process inference (MAGI) method for Bayesian parameter inference and trajectory reconstruction of ODE-based dynamical systems, focusing primarily on sparse and noisy data conditions. First, we introduce Pilot MAGI (pMAGI), a novel methodological upgrade on the base MAGI method that confers significantly-improved numerical stability, parameter inference, and trajectory reconstruction. Second, we demonstrate, for the first time to our knowledge, how one can combine MAGI-based methods with dynamical systems theory to provide probabilistic classifications of whether a system is stable or chaotic. Third, we demonstrate how pMAGI performs favorably in many settings against much more computationally-expensive and overparameterized methods. Fourth, we introduce Pilot MAGI Sequential Prediction (PMSP), a novel method building upon pMAGI that allows one to predict the trajectory of ODE-based dynamical systems multiple time steps into the future, given only sparse and noisy observations. We show that PMSP can output accurate future predictions even on chaotic dynamical systems and significantly outperform PINN-based methods. Overall, we contribute to the literature two novel methods, pMAGI and PMSP, that serve as Bayesian, uncertainty-quantified competitors to the Physics-Informed Neural Network.