Fourier Feature Networks Enhance PDE Solving Accuracy.

Solving Partial Differential Equations (PDEs) is a critical task across many scientific and engineering disciplines. This research proposes a new approach called Fourier Feature Networks (FENs), which are single-hidden-layer neural networks designed specifically for this purpose. Unlike some existing methods, FENs incorporate Fourier features, utilizing cosine, sine, or a combination of both, to construct a set of basis functions. Similar to Extreme Learning Machines (ELMs), FENs approximate the target function as a linear combination of these basis functions, with coefficients determined efficiently using the least squares method. A key distinction, however, is that FENs can achieve high-precision solutions without needing complex affine transformations on the input variables, which ELMs often rely on to boost their representational power. To assess their capabilities, the researchers optimized a scaling factor for the randomly initialized, fixed weights and biases within a predefined range. This allowed for a fair comparison against ELMs using various activation functions like sigmoid, tanh, and swish. Numerical experiments consistently demonstrated that FENs achieved superior accuracy compared to ELMs, highlighting their enhanced representational capacity and efficiency for solving PDEs.