PDE Operator Learning

PDE: Partial Differential Equation

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24 papers in the last four weeks, up 140% on the four weeks before. 0.2% of all new papers.

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CAFE+FNO: Fourier Kernel Generation via Multiplicative Feature Composition
Oct 7, 2026Hyungjoon Juen, Minwoo ShinFourier Neural Operator PDE Operator Learning
Department of Software, Yonsei University (Mirae Campus), Wonju 26493, Republic of Korea
Learning PDE solution operators with variable initial conditions via Latent Dynamics Networks
Oct 6, 2026Stefano Maria Pizzamiglio, Stefano Pagani, Francesco RegazzoniLatent Dynamics Modeling Meta-Learning
MOX Laboratory, Department of Mathematics, Politecnico di Milano, Italy
OCL-PDE: A Generative Framework for PDE Inverse Problems with Observation-Complementary Latents
Oct 5, 2026Ding Yang, Chuqi Chen, Chang Ma +1Physics-Informed Generative Modeling PDE Operator Learning
Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong · Department of Mathematics, University of Michigan, Ann Arbor, USA · Department of Materials, The Hong Kong University of Science and Technology, Hong Kong +1
Transmission Factors for Lossy Compression of PDE Training Data: Measuring What Reaches a Trained Operator
Oct 5, 2026Huy Hoang LeRate-Distortion Optimization PDE Operator Learning
Powermore Ltd., Da Nang, Vietnam
Poisson-GENERIC Neural Operators: Exact Metriplectic Structure in Function Space via Casimir Entropies
Oct 4, 2026Jason Sulskis, Sathya RaviDynamical Systems PDE Operator Learning
Georgia Tech Research Institute, Atlanta, GA, USA
Green-Routed Neural Operators:\Physics Determines Where the Network Reads
Oct 4, 2026Chenhao Si, Ming YanPDE Surrogate Modeling PDE Operator Learning
School of Data Science The Chinese University of Hong Kong, Shenzhen Shenzhen, China
METRO: Metric-Enhanced Token Routing Operator
Oct 4, 2026Nodens Koren, Thomas Hofmann, Georgios KissasMetric Learning PDE Operator Learning
Department of Computer Science ETH Zürich Zürich, Switzerland · Swiss Data Science Center ETH Zürich Zürich, Switzerland
Advectra: Asymmetric Latent Transport for Non-Stationary Physics
Oct 4, 2026Nodens Koren, Thomas Hofmann, Georgios KissasPDE Operator Learning Neural Operators
Department of Computer Science ETH Zürich Zürich, Switzerland · Swiss Data Science Center ETH Zürich Zürich, Switzerland
Learning PDE Dynamics between Submanifolds Using Green's Observation Operators
Oct 1, 2026Jan Tauberschmidt, Jephte Abijuru, Naukshatro Bose +5PDE Solving PDE Surrogate Modeling
German Research Center for Artificial Intelligence (DFKI), Kaiserslautern, Germany · Rheinland-Pfälzische Technische Universität (RPTU), Kaiserslautern, Germany · University of Manchester, Manchester, United Kingdom
Beyond Simulation: Retain-and-Repair Neural Operators for Real-World Adaptation
Sep 30, 2026Woojin Cho, Junghwan ParkSynthetic-to-Real Domain Adaptation PDE Operator Learning
TelePIX
Geometry-physics confounding impairs PDE learning across varying domains
Sep 29, 2026Yinghao Cheng, Gengxiang Chen, Xu Liu +6PDE Discovery PDE Operator Learning
College of Mechanical & Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China · School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast, United Kingdom · School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast, United Kingdom +3
Transolver- $σ$ : Joint Spectral-Physical Subspace Modeling for Neural PDE Solving
Sep 29, 2026Haonan Shangguan, Hang Zhou, Haixu Wu +3Fourier Neural Operator Neural PDE Solvers
School of Software, BNRist, Tsinghua University · MIT CSAIL
Neural Harmonic Measure Operator
Sep 28, 2026Jinjin He, Sinan Wang, Yuchen Sun +1Neural PDE Solvers PDE Solving
Georgia Institute of Technology
Theory Guided and Interpretable Neural Operator Design for Partial Differential Equation Learning
Sep 27, 2026Zeyuan Song, Zheyu JiangNeural PDE Solvers Neural Network Interpretability
Oklahoma State University Stillwater, OK 74078
Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies
Sep 27, 2026Xiang Li, Yue SongHodge Decomposition PDE Operator Learning
Tsinghua University · Beijing University of Chemical Technology
Elucidating the Conformal Structure of the Brinkman Penalisation Method for Geometry-Adapted, Structure-Preserving Operator Learning of Hamiltonian PDEs
Sep 24, 2026Teo Deveney, Baige Xu, Takaharu YaguchiPDE Operator Learning Neural Operators
Department of Computer Science, University of Bath, United Kingdom · Center for Advanced Intelligence Project, RIKEN, Japan · Kyushu University / Center for Advanced Intelligence Project, RIKEN, Japan
Physics and Data Driven Transformer-Mamba Framework for Flow Field
Sep 24, 2026Zhuo Zhang, Shun Zou, Canqun Yang +1PDE Surrogate Modeling PDE Operator Learning
College of Computer Science and Technology, National University of Defense Technology, Changsha, China · College of Artificial Intelligence, Nanjing Agricultural University, NanJing, China · National SuperComputer Center in Tianjin, Binhai New Area, Tianjin, China +1
MENO: Memory-Efficient Neural Operator
Sep 23, 2026Shengyang Xu, Weijun Zhang, Jun Hu +1PDE Surrogate Modeling PDE Operator Learning
School of Mathematical Sciences, Peking University, Beijing 100871, China · Chongqing Research Institute of Big Data, Peking University, Chongqing 401329, China · National Engineering Laboratory for Big Data Analysis and Applications, Peking University, Beijing 100871, China
Predicting Out-of-Distribution Generalization of Neural Operators via Observable Spectral Error Decomposition
Sep 20, 2026Hang-Cheng Dong, Pengcheng ChengOOD Generalization PDE Operator Learning
School of Instrumentation Science and Engineering, Harbin Institute of Technology, Harbin 150001, China · School of Mathematics, Jilin University, Changchun 130012, China
HiLNO: A Hierarchical Latent Neural Operator with Multi-Scale Supervision for PDEs on General Geometries
Sep 16, 2026Zhicheng Hu, Jiacheng Li, Min YangHierarchical Representation Learning Multi-Scale Feature Fusion
School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China · Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing, 211106, China · School of Mathematics and Information Sciences, Yantai University, Yantai, China
Lecture notes on Physics Informed Neural Networks, Neural Operators, and their applications
Sep 15, 2026Alessandro BombiniPDE Operator Learning Physics-Informed ML
Istituto Nazionale di Fisica Nucleare, Sezione di Firenze
Can Deep Learning Achieve Cross-Physics Mapping?
Sep 15, 2026Pengfei Zhu, Julien Lecompagnon, Mathias ZieglerPDE Operator Learning Neural Operators
Bundesanstalt für Materialforschung and -prüfung (BAM), 12205 Berlin, Germany
Stable by Construction: Variational Latent Markov Operators for Long-Horizon PDE Prediction
Sep 15, 2026Junyi Liao, Johann Guilleminot, Vahid TarokhVariational Autoencoders Latent Dynamics Modeling
†Department of Electrical and Computer Engineering, Duke University · ‡Department of Mechanical Engineering and Materials Science, Duke University
Single-condition neural solvers encode transferable response spaces for parametric differential equations
Sep 14, 2026Wenbo Cao, Weiwei ZhangPDE Solving PDE Surrogate Modeling
Institute of AI for Industries, Chinese Academy of Sciences, Nanjing 211135, China · Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China · School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China +2
Physics-Informed Conformal Prediction: Embedding PDE Consistency into Distribution-Free Uncertainty Quantification for Neural Operators
Sep 14, 2026Michael ChinFourier Neural Operator Uncertainty Quantification
Independent Researcher
DU-NO: A Parameter-Efficient Double U-Shaped Neural Operator for Phase-Resolving Wave Modeling
Sep 14, 2026Enrique Hernandez Noguera, Md Meftahul Ferdaus, Nathan Cooper +2U-Net PDE Operator Learning
Canizaro-Livingston Gulf States Center for Environmental Informatics, LSU New Orleans, New Orleans, LA, USA · U.S. Naval Research Laboratory, Stennis Space Center, MS, USA
Two-Scale Localized PCA-Net: Coarse-Global and Local-Residual Representations for Artifact-Reduced PDE Operator Learning
Sep 7, 2026Mrigank Dhingra, Jordan Stout, Omer SanPrincipal Component Analysis Reduced-Order Modeling
Department of Mechanical & Aerospace Engineering University of Tennessee Knoxville, Tennessee 37917, USA · Department of Mathematics and Statistics Boston University Boston, Massachusetts 02215, USA
Local gradient neural operator
Sep 7, 2026Baiming Zhang, Jinsong Tang, Ying Xu +2PDE Surrogate Modeling PDE Operator Learning
Zhejiang University · Nanjing University of Science and Technology · Hefei University of Technology
Equation Recast for Canonical Operator Learning Across Parametric PDEs
Sep 2, 2026Qiyun Cheng, Valentin Duruisseaux, Cesar F. Clauser +7PDE Operator Learning Operator Learning
1Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA, USA · 2Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA · Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA +1
VATO: A Vortex-Force-Aware Transformer Operator for Unsteady Separated Aerofoil Flows
Sep 1, 2026Xingxin Yang, Zhan Zhang, Yichen Li +1PDE Operator Learning Neural Operators
aDepartment of Engineering, King’s College London, London, WC2R 2LS, UK

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CAFE+FNO: Fourier Kernel Generation via Multiplicative Feature Composition

Hyungjoon Juen, Minwoo Shin

Department of Software, Yonsei University (Mirae Campus), Wonju 26493, Republic of Korea

Fourier Neural Operator PDE Operator Learning Fourier Feature Embeddings Compositional Representation Learning

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The Fourier Neural Operator (FNO) learns solution operators of partial differential equations (PDEs) through Fourier-space kernel parameterization, but frequency truncation can limit the learning of high-frequency variations. AM-FNO and SirenFNO generate kernels for all grid modes from spectral coordinates using shared networks, making coordinate encoding and generator design important. Recent work on implicit neural representations (INRs) has proposed constructing frequency interactions through explicit feature composition rather than relying on subsequent MLPs to form them implicitly. Building on this approach, we propose CAFE+FNO, which incorporates Content-Aware Frequency Encoding+ (CAFE+) into Fourier kernel generation. CAFE+ combines Fourier--Chebyshev features through parallel affine branches and a Hadamard product, forming interactions within and across the two feature families. A kernel MLP maps the resulting representation of each normalized spectral coordinate to a complex channel-mixing matrix. Each layer shares its generator across all stored modes, making the number of trainable parameters independent of the number of modes for a fixed architecture. We compare CAFE+FNO with existing FNO variants on five PDE benchmarks and conduct ablation studies on basis configuration, multiplicative composition, and bandwidth learnability. Code and experimental configurations are available at https://github.com/fabsk101/CAFEPlusFNO.git.

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