Efficient Learning of PDEs via Taylor Expansion and Sparse Decomposition
  into Value and Fourier Domains
Md Nasim Yexiang Xue
Accelerating the learning of Partial Differential Equations (PDEs) from
experimental data will speed up the pace of scientific discovery. Previous
randomized algorithms exploit sparsity in PDE updates for acceleration. However
such methods are applicable to a limited class of decomposable PDEs, which have
sparse features in the value domain. We propose Reel, which accelerates the
learning of PDEs via random projection and has much broader applicability. Reel
exploits the sparsity by decomposing dense updates into sparse ones in both the
value and frequency domains. This decomposition enables efficient learning when
the source of the updates consists of gradually changing terms across large
areas (sparse in the frequency domain) in addition to a few rapid updates
concentrated in a small set of "interfacial" regions (sparse in the value
domain). Random projection is then applied to compress the sparse signals for
learning. To expand the model applicability, Taylor series expansion is used in
Reel to approximate the nonlinear PDE updates with polynomials in the
decomposable form. Theoretically, we derive a constant factor approximation
between the projected loss function and the original one with poly-logarithmic
number of projected dimensions. Experimentally, we provide empirical evidence
that 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 original
size) with comparable quality as the non-compressed models.