  Understanding real-world dynamical phenomena remains a challenging task.
Across various scientific disciplines, machine learning has advanced as the
go-to 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, physics-informed 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 fully-connected
feed-forward neural network to explore a simple and illustrative model system,
the outbreak dynamics of COVID-19. Our Physics-Informed 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 self-adaptively 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.
