Neural Operators

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Neural operators learn maps from input fields, parameters, or geometries to output fields, often approximating the solution map of a partial differential equation. DeepONet uses branch and trunk networks, Fourier Neural Operators learn interactions in spectral space, and graph and transformer operators handle irregular meshes and complex geometries; physics-informed losses and structure-preserving designs add physical constraints. These models can make repeated simulation, forecasting, and inverse design much faster than running a numerical solver for every case. A central challenge is making predictions reliable across new resolutions, geometries, and physical regimes while preserving stability over long rollouts and matching classical solvers when high accuracy is required.

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313 papers52 papers in the last four weeks, up 24% on the four weeks before. 0.3% of all new papers.

UFO: A Domain-Unification-Free Operator Framework for Generalized Operator Learning
May 12, 2026Hanli Qiao, George Em Karniadakis, Muhammad MuniruzzamanCross-Domain Generalization PDE Operator Learning
Water and Mining Environment Unit, Geological Survey of Finland, Vuorimiehentie 5, Espoo, 02151, Finland · Division of Applied Mathematics, Brown University, 170 Hope Street, Providence, RI 02912, USA · Institute of Geosciences, University of Bonn, Kirschallee 1-3, Bonn, 53115, Germany
Neural-Schwarz Tiling for Geometry-Universal PDE Solving at Scale
May 12, 2026Paolo Secchi, Daniel S. Balint, Marco MauriziNeural PDE Solvers PDE Solving
Imperial College London · Italian Institute of Artificial Intelligence
Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
May 12, 2026Takashi Furuya, Ryo Ozawa, Jenn-Nan WangReaction-Diffusion Systems PDE Operator Learning
Doshisha University, RIKEN AIP · Tohoku University · National Taiwan University
NOFE - Neural Operator Function Embedding
May 12, 2026Lars Uebbing, Harald L. Joakimsen, Siyan Chen +6Inverse Dimensionality Reduction Sheaf Neural Networks
1UiT The Arctic University of Norway · 2Norwegian Computing Center · University of South Brittany +1
Compositional Neural Operators for Multi-Dimensional Fluid Dynamics
May 12, 2026Hamda Hmida, Hsiu-Wen Chang, Youssef MesriPDE Operator Learning Physics-Informed Neural Operators
Mines Paris - PSL University, Centre for Material Forming (CEMEF) · Mines Paris - PSL University, Centre for Robotics (CAOR)
Don't Fix the Basis -- Learn It: Spectral Representation with Adaptive Basis Learning for PDEs
May 11, 2026Xuxiang Zhao, Angelica I. Aviles-RiveroFourier Neural Operator Representation Learning
Qiuzhen College, Tsinghua University · YMSC, Tsinghua University
jNO: A JAX Library for Neural Operator and Foundation Model Training
May 11, 2026Leon Armbruster, Rathan Ramesh, Georg Kruse +1PDE Foundation Models PDE Operator Learning
Fraunhofer Institute for Integrated Systems and Device Technology IISB, Germany
HS-FNO: History-Space Fourier Neural Operator for Non-Markovian Partial Differential Equations
May 10, 2026Lennon J. ShikhmanFourier Neural Operator PDE Solving
College of Computing, Georgia Institute of Technology Atlanta, GA, United States
Bridging Spectral Operator Learning and U-Net Hierarchies: SpectraNet for Stable Autoregressive PDE Surrogates
May 9, 2026Enrique Hernández Noguera, Md Meftahul Ferdaus, Elias Ioup +2U-Net PDE Surrogate Modeling
University of New Orleans, New Orleans, LA, USA · Naval Research Laboratory, Stennis Space Center, MS, USA
CATO: Charted Attention for Neural PDE Operators
May 9, 2026Chun-Wun Cheng, Sifan Wang, Carola-Bibiane Schönlieb +1PDE Operator Learning Attention Mechanisms
DAMTP University of Cambridge · Institute for Foundations of Data Science Yale University · Yau Mathematical Sciences Center Tsinghua University
Neural Operators as Efficient Function Interpolators
May 8, 2026Vasilis Niarchos, Angelos Sirbu, Sokratis TrifinopoulosNeural Surrogate Modeling Fourier Neural Operator
Department of Physics, CCTP and ITCP, University of Crete, 71303, Greece · Theoretical Physics Department, CERN, Geneva, Switzerland · Physik-Institut, Universität Zürich, 8057 Zürich, Switzerland
QuadNorm: Resolution-Robust Normalization for Neural Operators
May 8, 2026Bum Jun Kim, Makoto Kawano, Yusuke Iwasawa +1Numerical Integration PDE Operator Learning
The University of Tokyo, Japan
One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators
May 7, 2026Panos Tsimpos, Edoardo Calvello, Ayoub Belhadji +1Amortized Inference Bayesian Inference
Operations Research Center MASSachusetts Institute of Technology Cambridge, MA 02139 · Department of Computing and Mathematical Sciences California Institute of Technology Pasadena, CA 91125 · Laboratory for Information and Decision Systems Center for Computational Science and Engineering MASSachusetts Institute of Technology Cambridge, MA 02139 +2
Do Neural Operators Forget Geometry? The Forgetting Hypothesis in Deep Operator Learning
May 7, 2026Yanming Xia, Angelica I. Aviles-RiveroNeural Network Generalization Neural Operators
Qiuzhen College, Tsinghua University, Beijing, China · Yau Mathematical Sciences Center, Tsinghua University, Beijing, China.
A Robust Foundation Model for Conservation Laws: Injecting Context into Flux Neural Operators via Recurrent Vision Transformers
May 6, 2026Taeyoung Kim, Joon-Hyuk KoPDE Solving Neural Operators
Center for AI and Natural Sciences Korea Institute for Advanced Study Seoul, South Korea 02455
Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations
May 6, 2026Zhangyong Liang, Huanhuan GaoNeural PDE Solvers PDE Solving
National Center for Applied Mathematics Tianjin University Tianjin, 300072, China · School of Mechanical and Aerospace Engineering, Jilin University Jilin University Changchun, 130025, China
Replay-Based Continual Learning for Physics-Informed Neural Operators
May 6, 2026Yizheng Wang, Mohammad Sadegh Eshaghi, Xiaoying Zhuang +2OOD Generalization Replay-Based Continual Learning
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China · Institute of Photonics, Department of Mathematics and Physics, Leibniz University Hannover, Germany · Institute of Structural Mechanics, Bauhaus-Universität Weimar, Marienstr. 15, D-99423 Weimar, Germany
GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains
May 5, 2026James Rowbottom, Elizabeth L. Baker, Nick Huang +3Graph Neural Operators Finite Element Method
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK · Department of Applied Mathematics and Computer Science, Technical University of Denmark, Denmark · Department of Mathematics, Simon Fraser University, Canada +1
Isotropic Fourier Neural Operators
May 4, 2026Michael F. StaddonFourier Neural Operator Neural PDE Solvers
Independent Researcher
ZNO: Stable Rational Neural Operators in the Z-Domain for Discrete-Time Dynamics
May 4, 2026Xianli Zhu, Jia YinSystem Identification Neural Operators
School of Mathematical Sciences Fudan University Shanghai, China
An ALE-Consistent Graph Neural Operator-Transformer Framework for Fluid-Structure Interaction
May 1, 2026Shihang Zhao, Martín Saravia, Haokui Jiang +2PDE Surrogate Modeling Graph Neural Operators
Institute for Ocean Engineering, Shenzhen International Graduate School, Tsinghua University, Shenzhen, Guangdong, 518055, China · Centro de Investigación en Mecánica Teórica y Aplicada, Universidad Tecnológica Nacional - CONICET, 11 de Abril 461, Bahía Blanca, Argentina
RETO: A Rotary-Enhanced Transformer Operator for High-Fidelity Prediction of Automotive Aerodynamics
Apr 30, 2026Bojun Zhang, Huiyu Yang, Yunpeng Wang +4Rotary Positional Embeddings Automotive Aerodynamics
Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, 518055, China · Shenzhen Key Laboratory of Complex Aerospace Flows, Southern University of Science and Technology,Shenzhen 518055, China · Ningbo Key Laboratory of Advanced Manufacturing Simulation, Eastern Institute of Technology, Ningbo, 315200, China +1
Learning Neural Operator Surrogates for the Black Hole Accretion Code
Apr 28, 2026Matthias Nägele, Cedric Bös, Chester Tan +3Computational Fluid Dynamics Transformer Neural Operator
Institute for Theoretical Physics and Astrophysics, Julius-Maximilians-Universität Würzburg, Würzburg, Germany · Chair for Machine Learning for Complex Networks, Julius-Maximilians-Universität Würzburg, Würzburg, Germany
Shearlet Neural Operators for Anisotropic-Shock-Dominated and Multi-scale parametric partial differential equations
Apr 28, 2026Fabio Pereira dos Santos, Julio de Castro Vargas Fernandes, Adriano Mauricio de Almeida CortesPDE Solving PDE Operator Learning
Coordination of Mathematical and Computational Methods National Laboratory of Scientific Computing Av. Getúlio Vargas, 333 - Quitandinha, Petrópolis - RJ, 25651-075 · Systems and Computational Engineering Program, COPPE/UFRJ, Federal University of Rio de Janeiro, P.O. Box 68542, Rio de Janeiro, RJ, 21941-909, Brasil
Droplet-LNO: Physics-Informed Laplace Neural Operators for Accurate Prediction of Droplet Spreading Dynamics on Complex Surfaces
Apr 22, 2026Ganesh Sahadeo Meshram, Partha Pratim Chakrabarti, Suman ChakrabortyPDE Operator Learning Physics-Informed Neural Operators
Department of Mechanical Engineering, IIT Kharagpur, Kharagpur, India · Department of Computer Science & Engineering, IIT Kharagpur, Kharagpur, India
AI models of unstable flow exhibit hallucination
Apr 22, 2026Ramdhan Wibawa, Birendra JhaScientific ML Neural Operators
Department of Chemical Engineering and Materials Science, University of Southern California, 925 Bloom Walk, Los Angeles, 90089, CA, USA.
Debiased neural operators for estimating functionals
Apr 21, 2026Konstantin Hess, Dennis Frauen, Niki Kilbertus +1Debiased ML Neural Operators
1LMU Munich · 2Munich Center for Machine Learning · 3TU Munich +1
Faster by Design: Interactive Aerodynamics via Neural Surrogates Trained on Expert-Validated CFD
Apr 20, 2026Nicholas Thumiger, Andrea Bartezzaghi, Mattia Rigotti +5PDE Surrogate Modeling Graph Neural Operators
IBM Research · Dallara Automobili
DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting
Apr 20, 2026Chih-Kang Huang, Ludovick Gagnon, Miha Založnik +1Physics-Informed Neural Operators Neural Operators
1Universit´e de Lorraine, CNRS, IJL, F-54000 Nancy, France · 2Universit´e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
Neural Shape Operator Surrogates -- Expression Rate Bounds
Apr 20, 2026Helmut Harbrecht, Christoph SchwabPDE Operator Learning Neural Network Approximation Theory

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UFO: A Domain-Unification-Free Operator Framework for Generalized Operator Learning

Hanli Qiao, George Em Karniadakis, Muhammad Muniruzzaman

Water and Mining Environment Unit, Geological Survey of Finland, Vuorimiehentie 5, Espoo, 02151, Finland · Division of Applied Mathematics, Brown University, 170 Hope Street, Providence, RI 02912, USA · Institute of Geosciences, University of Bonn, Kirschallee 1-3, Bonn, 53115, Germany

Cross-Domain Generalization PDE Operator Learning Operator Learning Neural Operators

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Neural operators have become an effective framework for learning mappings between function spaces, yet most existing architectures realize operators within a single representational domain, such as physical, spectral, or latent space. In this work, we introduce UFO (Domain-Unification-Free Operator), a cross-domain neural operator framework that realizes operators through adaptive, jointly conditioned interactions among representations defined on distinct domains. UFO enables discretization decoupling: the input function can be observed at resolutions or locations different from those used during training, while the solution can be queried at arbitrary output resolutions. Across four complementary benchmarks covering discontinuous inputs, irregular sampling with spectral mismatch, nonlinear dynamics, and stochastic high-frequency fields, UFO delivers accurate, robust, and physically coherent predictions under distribution shifts. These results establish cross-domain, phase-modulated realization as a powerful framework for discretization-decoupled neural operator learning.