A Bayesian Framework for learning governing Partial Differential Equation from Data
This work addresses the challenge of accurately identifying underlying PDEs in the presence of noise, which is incremental as it builds on existing machine learning approaches with a Bayesian framework.
The authors tackled the problem of discovering partial differential equations (PDEs) from noisy data by proposing a new method that combines variational Bayes and sparse linear regression, demonstrating its efficacy on examples like Burgers and wave equations.
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.