PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data
This addresses the challenge of PDE discovery for scientific data analysis, offering a robust method for realistic scenarios, though it appears incremental as it builds on existing deep learning approaches for PDE identification.
The paper tackles the problem of discovering governing partial differential equations (PDEs) from noisy, limited data, introducing PDE-LEARN, a deep learning algorithm that identifies PDEs directly from such measurements, demonstrating efficacy on several examples.
In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational Neural Network, $U$, to approximate the system response function and a sparse, trainable vector, $ξ$, to characterize the hidden PDE that the system response function satisfies. Our approach couples the training of $U$ and $ξ$ using a loss function that (1) makes $U$ approximate the system response function, (2) encapsulates the fact that $U$ satisfies a hidden PDE that $ξ$ characterizes, and (3) promotes sparsity in $ξ$ using ideas from iteratively reweighted least-squares. Further, PDE-LEARN can simultaneously learn from several data sets, allowing it to incorporate results from multiple experiments. This approach yields a robust algorithm to discover PDEs directly from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by identifying several PDEs from noisy and limited measurements.