aiqbench analysis
This research introduces SANO, a spatially adaptive neural operator that utilizes hypernetworks to generate location-dependent parameters for solving heterogeneous partial differential equations. By replacing shared parameterization with a continuous field of local operators, the method effectively mitigates underfitting in high-gradient regions and complex geometries. The approach demonstrates consistent performance gains over existing baselines across 1D, 2D, and 3D benchmarks, advancing the frontier of physics-informed machine learning.
Hypernetwork-Parameterized Spatially Adaptive Neural Operators for PDE Learning
arXiv:2609.20309v1 Announce Type: new 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.
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.
Machine Learning (cs.LG)
arXiv:2609.20309 [cs.LG]
(or arXiv:2609.20309v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.20309 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 259 words
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