A Bayesian Framework for learning governing Partial Differential
  Equation from Data
Kalpesh More Tapas Tripura Rajdip Nayek Souvik Chakraborty
The discovery of partial differential equations (PDEs) is a challenging task
that involves both theoretical and empirical methods. Machine learning
approaches have been developed and used to solve this problem; however, it is
important to note that existing methods often struggle to identify the
underlying equation accurately in the presence of noise. In this study, we
present a new approach to discovering PDEs by combining variational Bayes and
sparse linear regression. The problem of PDE discovery has been posed as a
problem to learn relevant basis from a predefined dictionary of basis
functions. To accelerate the overall process, a variational Bayes-based
approach for discovering partial differential equations is proposed. To ensure
sparsity, we employ a spike and slab prior. We illustrate the efficacy of our
strategy in several examples, including Burgers, Korteweg-de Vries, Kuramoto
Sivashinsky, wave equation, and heat equation (1D as well as 2D). Our method
offers a promising avenue for discovering PDEs from data and has potential
applications in fields such as physics, engineering, and biology.