# A Library for Learning Neural Operators

arXiv: https://arxiv.org/abs/2412.10354 · HTML: https://mlsanity.com/paper/2412.10354/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 27 references that are in the map.
- First posted on arXiv 13 Dec 2024.

## Builds on (most cited first, 18 of 18)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [TensorLy: Tensor Learning in Python](https://mlsanity.com/paper/1610.09555/) (2016)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction](https://mlsanity.com/paper/2402.01960/) (2024)
- [Incremental Spatial and Spectral Learning of Neural Operators for Solving Large-Scale PDEs](https://mlsanity.com/paper/2211.15188/) (2022)
- [Guaranteed Approximation Bounds for Mixed-Precision Neural Operators](https://mlsanity.com/paper/2307.15034/) (2023)
- [Tipping Point Forecasting in Non-Stationary Dynamics on Function Spaces](https://mlsanity.com/paper/2308.08794/) (2023)

## Built on by (most cited first, 30 of 38)

- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Towards a Physics Foundation Model](https://mlsanity.com/paper/2509.13805/) (2025)
- [Flow Straight and Fast in Hilbert Space: Functional Rectified Flow](https://mlsanity.com/paper/2509.10384/) (2025)
- [SMART: Scalable Mesh-free Aerodynamic Simulations from Raw Geometries using a Transformer-based Surrogate Model](https://mlsanity.com/paper/2601.18707/) (2026)
- [One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators](https://mlsanity.com/paper/2605.06873/) (2026)
- [Toward Data-Driven Surrogates of the Solar Wind with Spherical Fourier Neural Operator](https://mlsanity.com/paper/2511.22112/) (2025)
- [Provable Mixed-Noise Learning with Flow-Matching](https://mlsanity.com/paper/2508.18122/) (2025)
- [Efficient Generative Transformer Operators For Million-Point PDEs](https://mlsanity.com/paper/2512.04974/) (2025)
- [Residual-augmented flow matching operators for probabilistic partial differential equations](https://mlsanity.com/paper/2512.12749/) (2025)
- [Hypothesis-driven construction of mesoscopic dynamics](https://mlsanity.com/paper/2605.16211/) (2026)
- [Iterative Refinement Neural Operators are Learned Fixed-Point Solvers: A Principled Approach to Spectral Bias Mitigation](https://mlsanity.com/paper/2605.24041/) (2026)
- [Thompson Sampling in Function Spaces via Neural Operators](https://mlsanity.com/paper/2506.21894/) (2025)
- [Physics-Informed Neural Operators for Cardiac Electrophysiology](https://mlsanity.com/paper/2511.08418/) (2025)
- [Autoregressive Surrogate Modeling of the Solar Wind with Spherical Fourier Neural Operator](https://mlsanity.com/paper/2511.20830/) (2025)
- [Hybrid operator learning of wave scattering maps in high-contrast media](https://mlsanity.com/paper/2602.11197/) (2026)
- [Equation Recast for Canonical Operator Learning Across Parametric PDEs](https://mlsanity.com/paper/2609.02982/) (2026)
- [Preconditioned Physics-Informed Neural Operator Training](https://mlsanity.com/paper/2609.36216/) (2026)
- [Translation Invariance of Neural Operators for the FitzHugh-Nagumo Model](https://mlsanity.com/paper/2603.17523/) (2026)
- [Do Neural Operators Forget Geometry? The Forgetting Hypothesis in Deep Operator Learning](https://mlsanity.com/paper/2605.05862/) (2026)
- [QuadNorm: Resolution-Robust Normalization for Neural Operators](https://mlsanity.com/paper/2605.07375/) (2026)
- [Neural Operators as Efficient Function Interpolators](https://mlsanity.com/paper/2605.07792/) (2026)
- [Fourier Neural Operators with rank-1 lattice points and hyperbolic cross](https://mlsanity.com/paper/2606.08871/) (2026)
- [OperatorSHAP: Fast and Accurate Shapley Value Estimation for Neural Operators](https://mlsanity.com/paper/2606.28065/) (2026)
- [Higher-Order Fourier Neural Operator: Explicit Mode Mixer for Nonlinear PDEs](https://mlsanity.com/paper/2606.28122/) (2026)
- [Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data](https://mlsanity.com/paper/2607.02203/) (2026)
- [Neural Operator Surrogates for Two-Dimensional Neutron Flux Estimation](https://mlsanity.com/paper/2607.19388/) (2026)
- [DFSC: Error-Controlled Differentiable Mittag-Leffler Propagation for Fractional Scientific Machine Learning](https://mlsanity.com/paper/2607.29038/) (2026)
- [ADEx-FNO: A Unified Ambient-Domain Framework for Fourier Neural Operators on Varying Geometries](https://mlsanity.com/paper/2608.08608/) (2026)
- [Distillation of Foundation Models for Time-dependent PDEs](https://mlsanity.com/paper/2608.11937/) (2026)

## Abstract (arXiv)

We present NeuralOperator, an open-source Python library for operator learning. Neural operators generalize neural networks to maps between function spaces instead of finite-dimensional Euclidean spaces. They can be trained and inferenced on input and output functions given at various discretizations, satisfying a discretization convergence properties. Part of the official PyTorch Ecosystem, NeuralOperator provides all the tools for training and deploying neural operator models, as well as developing new ones, in a high-quality, tested, open-source package. It combines cutting-edge models and customizability with a gentle learning curve and simple user interface for newcomers.