Physics-based discovery of governing equations from scarce and noisy data

Discovering the governing equations of physical systems from data is crucial for our deep understanding and effective modeling of such systems. In this paper, we propose a framework that integrates physics-informed neural networks (PINNs) with sparse regression to discover partial differential control equations from scarce and noisy data. Specifically, the framework effectively reduces the size of the candidate function library through dimensional verification while leveraging the powerful nonlinear fitting capabilities and automatic differentiation features of deep neural networks to model physical systems and compute candidate functions. We investigate and validate the effectiveness and robustness of the proposed method in discovering equations from data using three examples: the Burgers equation, the heat equation, and the Navier-Stokes vorticity equation. Comparisons with other methods that combine PINNs with sparse regression based on overcomplete libraries demonstrate that our approach achieves better discovery accuracy and smaller prediction errors. Additionally, we discuss various details of the framework, such as the process of discovering equations and the influence of measurement points and residual points on the results.