Legendre polynomial anti-derivative layer for compact physics-informed neural networks and operator learning with high-frequency spectral expressivity

Abstract

Physics-informed neural networks (PINNs) with compact architectures often struggle to represent oscillatory and multiscale solutions when the representations formed by their hidden layers are insufficiently expressive. Here, we introduce a Legendre polynomial anti-derivative (LPA) layer as a structured readout activation that enriches the final solution representation without increasing network width or depth. The LPA maps the latent representation through a finite-dimensional Legendre polynomial basis whose coefficients are analytically assembled from trainable panel strengths. Its retained order controls the oscillatory capacity and the highest non-zero derivative order of the readout with respect to its latent argument, while spatial derivatives of the composed network remain generally non-zero. On the two-dimensional Helmholtz problem, LPA improves compact tanh PINNs by one to two orders of magnitude and, with the retained order fixed at three, provides additional gains on Fourier-feature and less-aligned sinusoidal trunks, while its marginal benefit is reduced when the upstream representation is already highly expressive. On diffusion–reaction and Kovasznay-flow problems, the baseline and LPA models attain comparable accuracy. When incorporated into a physics-informed deep operator network, LPA improves the accuracy of the velocity components at the two most compact widths, while at larger widths the ordering reverses or becomes mixed. These results identify structured readout activation as an effective means of increasing the representational efficiency of compact physics-informed models, particularly for solutions with significant high-frequency content.