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.

翻译：从实验数据中加速偏微分方程（PDE）的学习将提升科学发现的效率。先前的随机化算法利用PDE更新过程中的稀疏性实现加速，然而此类方法仅适用于可分解的PDE子类，且依赖值域中的稀疏特征。我们提出Reel方法，通过随机投影加速PDE学习，并具有更广泛的适用性。Reel通过将稠密更新分解为值域与频域中的稀疏分量，利用稀疏性进行学习。该分解使得当更新源包含大面积平缓变化的项（在频域中稀疏）以及集中在少量"界面"区域的快速更新（在值域中稀疏）时，仍能实现高效学习。随后应用随机投影压缩稀疏信号以支持学习。为扩展模型适用性，Reel采用泰勒级数展开，将非线性PDE更新近似为可分解形式的多项式。理论上，我们推导出投影损失函数与原损失函数之间在仅需多对数数量投影维度下的常数因子近似保证。实验上，我们提供实证证据表明：当数据压缩至原始尺寸的1%时，所提出的Reel方法可使PDE模型学习速度提升（训练时间减少70-98%），且模型质量与非压缩模型相当。