Bayesian PhysicsInformed Neural Networks for realworld nonlinear dynamical systems
Abstract
Understanding realworld dynamical phenomena remains a challenging task. Across various scientific disciplines, machine learning has advanced as the goto technology to analyze nonlinear dynamical systems, identify patterns in big data, and make decision around them. Neural networks are now consistently used as universal function approximators for data with underlying mechanisms that are incompletely understood or exceedingly complex. However, neural networks alone ignore the fundamental laws of physics and often fail to make plausible predictions. Here we integrate data, physics, and uncertainties by combining neural networks, physicsinformed modeling, and Bayesian inference to improve the predictive potential of traditional neural network models. We embed the physical model of a damped harmonic oscillator into a fullyconnected feedforward neural network to explore a simple and illustrative model system, the outbreak dynamics of COVID19. Our PhysicsInformed Neural Networks can seamlessly integrate data and physics, robustly solve forward and inverse problems, and perform well for both interpolation and extrapolation, even for a small amount of noisy and incomplete data. At only minor additional cost, they can selfadaptively learn the weighting between data and physics. Combined with Bayesian Neural Networks, they can serve as priors in a Bayesian Inference, and provide credible intervals for uncertainty quantification. Our study reveals the inherent advantages and disadvantages of Neural Networks, Bayesian Inference, and a combination of both and provides valuable guidelines for model selection. While we have only demonstrated these approaches for the simple model problem of a seasonal endemic infectious disease, we anticipate that the underlying concepts and trends generalize to more complex disease conditions and, more broadly, to a wide variety of nonlinear dynamical systems.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 arXiv:
 arXiv:2205.08304
 Bibcode:
 2022arXiv220508304L
 Keywords:

 Computer Science  Machine Learning;
 Mathematics  Dynamical Systems;
 Nonlinear Sciences  Chaotic Dynamics;
 62Mxx;
 70Kxx;
 G.3;
 J.3