Spectral Allocation and Routing for Physics-Informed Learning

Physics-informed neural networks (PINNs) often struggle with spatially varying high-frequency solutions of partial differential equations (PDEs) due to spectral bias. Existing approaches, such as Fourier features, periodic activations, multiscale architectures, and adaptive Fourier representations, mitigate this bottleneck but often rely on either globally learned spectral representations or locally adapting predefined spectral bases. We introduce Spectral Allocation and Routing (SAR), an adaptive mechanism that learns a global set of Fourier frequencies and controls their local contributions through continuous, frequency-conditioned spatial routing. This formulation enables direct inspection of the learned spectral content and provides a straightforward way to incorporate spectral inductive biases through structured support parameterizations. Controlled experiments on nonstationary analytical targets investigate spectral discovery and local routing, while neural tangent kernel (NTK) analysis examines their effects on learning dynamics. We further evaluate SAR on heterogeneous Helmholtz problems, where it substantially improves reconstruction accuracy over unrouted representations and achieves low errors across spatially varying and localized multiscale wavefields.