# Hypernetwork-Parameterized Spatially Adaptive Neural Operators for PDE Learning

arXiv: https://arxiv.org/abs/2609.20309 · HTML: https://mlsanity.com/paper/2609.20309/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 44 references that are in the map.
- First posted on arXiv 31 Jul 2026.

## Builds on (most cited first, 19 of 19)

- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Learning to Accelerate Partial Differential Equations via Latent Global Evolution](https://mlsanity.com/paper/2206.07681/) (2022)
- [Learning Neural PDE Solvers with Parameter-Guided Channel Attention](https://mlsanity.com/paper/2304.14118/) (2023)
- [HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork](https://mlsanity.com/paper/2312.15949/) (2023)
- [Multi-scale Time-stepping of Partial Differential Equations with Transformers](https://mlsanity.com/paper/2311.02225/) (2023)
- [Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods](https://mlsanity.com/paper/2410.06820/) (2024)
- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions](https://mlsanity.com/paper/2509.05117/) (2025)
- [Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings](https://mlsanity.com/paper/2602.16193/) (2026)
- [Geometric Neural Operators via Lie Group-Constrained Latent Dynamics](https://mlsanity.com/paper/2602.16209/) (2026)
- [Autoregression-Free Neural Operators for Time-Dependent PDEs](https://mlsanity.com/paper/2605.25413/) (2026)

## Abstract (arXiv)

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.