Discovering governing partial differential equations from noisy data

Abstract

Partial differential equations (PDEs) derived from first principles knowledge have been indispensable tools for modelling many physical and chemical systems. However, the presence of complex terms in the PDEs may render traditional first principles modelling techniques inadequate. In such cases, data-driven methods such as PDE-FIND can be used to extract PDEs from the spatiotemporal measurements of the system. However, the PDE-FIND algorithm is sensitive to noise. Furthermore, while PDE-discovery models specifically developed for noisy data have been proposed, these models work best for low noise levels. Moreover, most of these models fail to discover the heat equation. We propose a smoothing-based approach for discovering the PDEs from noisy measurements. The framework is broadly data-driven, and its performance can be further improved by incorporating first principles knowledge (such as the order of the system). Our proposed algorithm effectively extracts partial differential equations (PDEs) from measurements with a low signal-to-noise ratio (SNR), outperforming existing techniques. Additionally, we have demonstrated the effectiveness of our algorithm in a real system (where collinear terms occur in the library) by using a new benchmark metric.