PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data
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In this paper, we introduce PDE-LEARN, a novel PDE discovery 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.
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