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In this paper, we introduce PDE-LEARN, a novel deep learning algorithm thatcan identify governing partial differential equations (PDEs) directly fromnoisy, limited measurements of a physical system of interest. PDE-LEARN uses aRational Neural Network, $U$, to approximate the system response function and asparse, trainable vector, $\xi$, to characterize the hidden PDE that the systemresponse function satisfies. Our approach couples the training of $U$ and $\xi$using a loss function that (1) makes $U$ approximate the system responsefunction, (2) encapsulates the fact that $U$ satisfies a hidden PDE that $\xi$characterizes, and (3) promotes sparsity in $\xi$ using ideas from iterativelyreweighted least-squares. Further, PDE-LEARN can simultaneously learn fromseveral data sets, allowing it to incorporate results from multipleexperiments. This approach yields a robust algorithm to discover PDEs directlyfrom realistic scientific data. We demonstrate the efficacy of PDE-LEARN byidentifying several PDEs from noisy and limited measurements.

PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data