The paper introduces Hypernetwork-Parameterized Spatially Adaptive Neural Operators (SANO) for learning partial differential equations (PDEs) with spatial heterogeneity. SANO replaces spatially shared parameterization with a continuous field of location-dependent operator parameters, generated by a coordinate-conditioned hypernetwork and interpolated via a Hyper-Neural Element mechanism. Experiments on 1‑, 2‑, and 3‑dimensional PDEs and perforated-domain elliptic benchmarks demonstrate that SANO consistently outperforms existing neural‑operator, hypernetwork‑based, and physics‑informed baselines.
arXiv:2606. 11963v1 Announce Type: new Abstract: Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space.
arXiv:2602. 11626v3 Announce Type: replace-cross Abstract: Learning solution operators on arbitrary geometries remains a central challenge in scientific machine learning, especially for many-query simulation, physics-informed learning, and evolving geometries requiring accurate, geometry-aware predictions at arbitrary spatial locations.
arXiv:2609.38348v1 Announce Type: new Abstract: Many forms of data, including physical fields, geometric shapes, and visual signals, are naturally described by functions over continuous domains but a...
arXiv:2608. 06894v1 Announce Type: new Abstract: Neural operators have become a central tool for solving partial differential equations (PDEs), with spectral operators offering efficient global mixing across spatial locations.
arXiv:2608. 11237v1 Announce Type: new Abstract: Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs).
The paper introduces the Cluster Attention Neural Operator (CANO), a neural operator that uses a novel cross‑attention mechanism to dynamically cluster queries while keeping full‑resolution keys and values. This design eliminates the slice compression and weight‑sharing limitations of previous Transformer‑based operators, maintaining fast computation and global interactions. Experiments on a range of fluid and solid dynamics benchmarks—including Navier‑Stokes, Airfoil, Plasticity, Pipe Turbulence, and Composites—show that CANO achieves lower errors than existing baselines and demonstrates strong geometric adaptability and temporal consistency.
HiLNO is a hierarchical latent neural operator that builds a fine‑to‑coarse‑to‑fine latent space and incorporates multi‑scale supervision and anisotropic Gaussian attention to preserve spatial information in PDE solutions with multiscale structures. The hierarchy reduces information loss during compression, while multi‑scale supervision aligns intermediate predictions with downsampled targets, and anisotropic attention facilitates feature transfer across scales. Experiments on standard PDE benchmarks and a large‑scale automotive aerodynamics task show that HiLNO achieves competitive accuracy while cutting parameter count by 84.4% and FLOPs by 69.2% compared with LinearNO, and it generalizes effectively to unseen spatial resolutions.
arXiv:2609.07814v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) struggle on PDEs whose governing physics varies across the domain. We trace this to a structural property of s...
The paper investigates how different attention mechanisms affect the performance of DeepONet neural operators. Five variants—varying in cross‑attention, self‑attention, tokenization, and attention depth—are trained in both data‑driven and physics‑informed settings on one‑ and two‑dimensional PDE benchmarks. Results show that per‑sensor tokenization with cross‑attention consistently reduces error, while branch self‑attention helps only in complex spatial problems, and deeper cross‑attention yields diminishing returns with higher cost.
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
arXiv:2608. 09764v1 Announce Type: cross Abstract: Transformer-based neural operators have achieved substantial progress in solving Partial Differential Equations (PDEs) by projecting spatial observations into compact latent tokens and learning physical interactions in latent spaces.
arXiv:2608.30328v1 Announce Type: new Abstract: Classical numerical solvers for partial differential equations (PDEs) are computationally expensive to solve repeatedly across varying initial conditio...