Hypernetwork-Parameterized Spatially Adaptive Neural Operators for PDE Learning

Abstract

Spatially heterogeneous partial differential equations (PDEs) exhibit location-dependent dynamics arising from variations in geometry and physical coefficients. Existing neural operators improve localized modeling through multiscale features, attention mechanisms, or domain decomposition, yet their update rules often remain spatially shared. Hypernetwork-based methods adapt parameters across PDE instances but typically generate only one global parameterization per instance. Consequently, shared operators may underfit boundaries and high-gradient regions, with these localized errors accumulating during autoregressive rollout. We propose a spatially adaptive neural operator (SANO), which replaces this spatially shared parameterization with a spatially continuous field of location-dependent operator parameters. SANO uses Fourier-encoded coordinates and a coordinate-conditioned hypernetwork to generate spatial operator-conditioning codes at sampling points. A Hyper-Neural Element (HNE) mechanism interpolates these codes within local subregions, coupling neighboring operators while allowing their update rules to vary across space, and partition-of-unity weights assemble the overlapping local predictions. Experiments on one-, two-, and three-dimensional PDEs and two perforated-domain elliptic benchmarks show that SANO consistently outperforms competitive neural-operator, hypernetwork-based, and physics-informed baselines.