🎉 Hyperbolic Neural Operator (HNO) has been accepted to ICML 2026.
We present HNO, a neural operator that learns near-far interaction routing with stabilized Lorentz-hyperbolic distance kernels. HNO gives tokens a learned scale coordinate, enabling compact hierarchical PDE surrogate modeling across regular grids, structured meshes, point clouds, and large-scale CFD.
Key contributions include:
- Hyperbolic routing kernel: replaces dot-product token mixing with stabilized hyperbolic-distance attention.
- Near-far physical organization: learns FMM-inspired local-detail and far-field-summary structure without hand-built trees.
- Broad PDE and CFD validation: includes PDEBench tasks plus AirfRANS and ShapeNetCar large-scale unstructured meshes.
- Abstract and benchmark results
- Full paper PDF · Full-text HTML · Plain text
- Markdown research summary · BibTeX · Structured metadata
Neural operators learn solution operators for parametric PDE families, mapping coefficients, forcing fields, or geometric inputs to full solution fields and thereby accelerating scientific computation. Transformer-based architectures offer strong flexibility on irregular domains, but dense dot-product attention often allocates pairwise scoring uniformly across token pairs, neglecting that far-field interactions in many discretized PDE kernels are numerically compressible. To address this mismatch, we draw inspiration from classical fast solvers that exploit hierarchical near–far organization. We further observe that embedding such tree-structured hierarchies in Euclidean space incurs inherent distortion, whereas hyperbolic space naturally accommodates exponential branching. Consequently, we propose Hyperbolic Neural Operator (HNO), which leverages intrinsic hyperbolic geometry to instantiate a continuous Gibbs kernel based on stabilized geodesic distances on the Lorentz hyperboloid. This design imposes a geometric inductive bias for learnable multi-scale near–far routing within a unified attention mechanism. Empirically, HNO achieves the lowest error among the evaluated methods on six PDE benchmarks and two large-scale unstructured CFD tasks, reducing the mean relative ℓ2 error by up to 40% in the best evaluated setting. Code is available in the GitHub repository.
Hyperbolic Neural Operator (HNO) is an ICML 2026 method for learning solution operators of parametric partial differential equations. It maps input fields or geometric descriptors to solution fields. HNO uses stabilized geodesic distances on the Lorentz hyperboloid to construct a Gibbs attention kernel. This geometry provides an inductive bias for hierarchical near–far interaction routing, inspired by the organization of classical fast solvers. Here, hyperbolic refers to the learned representation geometry; the evaluated tasks span multiple PDE families. Far-field compressibility refers to numerical or low-rank structure in interactions. The paper evaluates HNO on Elasticity, Navier–Stokes, Darcy, Plasticity, Airfoil and Pipe, plus the AirfRANS and ShapeNet Car CFD benchmarks with approximately 32,000 mesh nodes per sample. The released implementations use geometry-specific tokenization, including patch-based grid models and summary-token processing for point clouds. The paper also examines hierarchical tree-kernel fitting, attention locality, and Darcy ablations. HNO is relevant to research on efficient neural operators, non-Euclidean attention, multiscale physical interactions, and PDE surrogates on irregular meshes. Reported accuracy and efficiency values describe the paper’s evaluated protocols.
python -m venv .venv_pdebench
source .venv_pdebench/bin/activate
python -m pip install --upgrade pip
python -m pip install -r requirements_pdebench.txt
bash scripts/smoke_test.shRun one task:
python -m pdebench.scripts.train_darcy --data_path <DARCY_DATA_DIR>pdebench/ HNO models, configs, and PDEBench training scripts
large_scale/ AirfRANS and ShapeNetCar code
scripts/ setup, smoke-test, and run wrappers
docs/ project website source
figures/ README figures
Datasets, checkpoints, logs, and generated caches are not included.
@inproceedings{hno2026,
title = {Hyperbolic Neural Operator},
author = {Pei, Jieyuan and Li, Zhuoxuan and Li, Wei and Zhang, Haobo and Jiang, Jiawei and Zheng, Jianwei},
booktitle = {Proceedings of the 43rd International Conference on Machine Learning},
series = {Proceedings of Machine Learning Research},
volume = {306},
publisher = {PMLR},
year = {2026},
url = {https://icml.cc/virtual/2026/poster/65554}
}MIT License. See LICENSE.