Inferring the Langevin Equation with Uncertainty via Bayesian Neural
  Networks
Youngkyoung Bae Seungwoong Ha Hawoong Jeong
Pervasive across diverse domains, stochastic systems exhibit fluctuations in
processes ranging from molecular dynamics to climate phenomena. The Langevin
equation has served as a common mathematical model for studying such systems,
enabling predictions of their temporal evolution and analyses of thermodynamic
quantities, including absorbed heat, work done on the system, and entropy
production. However, inferring the Langevin equation from observed trajectories
remains challenging, particularly for nonlinear and high-dimensional systems.
In this study, we present a comprehensive framework that employs Bayesian
neural networks for inferring Langevin equations in both overdamped and
underdamped regimes. Our framework first provides the drift force and diffusion
matrix separately and then combines them to construct the Langevin equation. By
providing a distribution of predictions instead of a single value, our approach
allows us to assess prediction uncertainties, which can prevent potential
misunderstandings and erroneous decisions about the system. We demonstrate the
effectiveness of our framework in inferring Langevin equations for various
scenarios including a neuron model and microscopic engine, highlighting its
versatility and potential impact.