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Accelerating the learning of Partial Differential Equations (PDEs) fromexperimental data will speed up the pace of scientific discovery. Previousrandomized algorithms exploit sparsity in PDE updates for acceleration. Howeversuch methods are applicable to a limited class of decomposable PDEs, which havesparse features in the value domain. We propose Reel, which accelerates thelearning of PDEs via random projection and has much broader applicability. Reelexploits the sparsity by decomposing dense updates into sparse ones in both thevalue and frequency domains. This decomposition enables efficient learning whenthe source of the updates consists of gradually changing terms across largeareas (sparse in the frequency domain) in addition to a few rapid updatesconcentrated in a small set of "interfacial" regions (sparse in the valuedomain). Random projection is then applied to compress the sparse signals forlearning. To expand the model applicability, Taylor series expansion is used inReel to approximate the nonlinear PDE updates with polynomials in thedecomposable form. Theoretically, we derive a constant factor approximationbetween the projected loss function and the original one with poly-logarithmicnumber of projected dimensions. Experimentally, we provide empirical evidencethat our proposed Reel can lead to faster learning of PDE models (70-98%reduction in training time when the data is compressed to 1% of its originalsize) with comparable quality as the non-compressed models.

Efficient Learning of PDEs via Taylor Expansion and Sparse Decomposition into Value and Fourier Domains