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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Loss-Conditional PINNs for Parametric PDE Families
Jun 3, 2026Anna Lazareva, Alexander TarakanovPDE Solving PDE Operator Learning
Faculty of Computer Science HSE University, Moscow, Russia · VK and HSE University Moscow, Russia
Let There Be Light: Reflection, Refraction and Scattering for Neural Operators
Jun 2, 2026Keke Wu, Yixuan Zhang, Jingrun ChenPDE Operator Learning Neural Operators
Suzhou Institute for Advanced Research, University of Science and Technology of China, Suzhou 215123, China · School of Artificial Intelligence and Data Science, University of Science and Technology of China, Hefei 230026, China · Suzhou Big Data & AI Research and Engineering Center, Suzhou 215123, China +2
EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs
Jun 2, 2026Sungwon Kim, Juho Song, Seungmin Shin +3Fourier Neural Operator PDE Surrogate Modeling
Graduate School of Data Science, KAIST, Daejeon, Republic of Korea · Mathematical Sciences, KAIST, Daejeon, Republic of Korea · LG Electronics, Pyeong-taek, Republic of Korea +1
Spectral Audit of In-Context Operator Networks
Jun 1, 2026Zhiwei Gao, Liu Yang, George Em KarniadakisPDE Operator Learning Operator Learning
Division of Applied Mathematics, Brown University, Providence, RI, USA · Department of Mathematics, National University of Singapore, Singapore
Functional Attention: From Pairwise Affinities to Functional Correspondences
May 29, 2026Jiefang Xiao, Maolin Gao, Simon Weber +2PDE Operator Learning Attention Mechanisms
Technical University of Munich, Germany · Munich Center for Machine Learning (MCML), Germany · PIXL, Department of Computer Science, University of Oxford, United Kingdom +1
LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition
May 29, 2026Jeongun Ha, Sanga Yoon, Donghun LeeDynamical Systems Neural PDE Solvers
Department of Mathematics, Korea University, 145 Anam-ro Seongbuk-gu Seoul, Republic of Korea.
Semigroup Consistency as a Diagnostic for Learned Physics Simulators
May 25, 2026Lennon J. ShikhmanNeural Surrogate Modeling PDE Operator Learning
Georgia Institute of Technology
Small Models, Strong Priors: Architectural Inductive Bias for Parameter-Efficient Neural PDE Solvers
May 25, 2026Shyam Sankaran, Hanwen Wang, Paris PerdikarisNeural PDE Solvers PDE Operator Learning
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania · Graduate Program in Applied Mathematics and Computational Science, University of Pennsylvania
Autoregression-Free Neural Operators for Time-Dependent PDEs
May 25, 2026Jiaquan Zhang, Caiyan Qin, Haoyu Bian +7Latent Dynamics Modeling PDE Operator Learning
School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China · School of Robotics and Advanced Manufacture, Harbin Institute of Technology, Shenzhen, Guangdong 518055, China · School of Mathematical Sciences, Capital Normal University, Beijing 100048, China +3
IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients
May 24, 2026Shan Zhong, George BirosNeural PDE Solvers PDE Solving
Oden Institute for Computational Science and Engineering, The University of Texas at Austin, 201 E 24th St, Austin, 78712, TX, USA · Walker Department of Mechanical Engineering, The University of Texas at Austin, 3012026 E Dean Keeton St, Austin, 78712, TX, USA
Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach
May 22, 2026Qian Zhang, George Em KarniadakisPDE Operator Learning Operator Learning
Division of Applied Mathematics, Brown University
IKNO: Infinite-order Kernel Neural Operators
May 21, 2026Pengyuan Zhu, Ivor W. Tsang, Yueming LyuPDE Operator Learning Neural Operators
Nanyang Technological University · Centre for Frontier AI Research(CFAR), Agency for Science, Technology and Research (A*STAR)
Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions
May 19, 2026Ha Dang, Sebastian Schmidt, Juergen HesserPDE Operator Learning Deep Operator Networks
Mannheim Institute for Intelligent Systems in Medicine, Heidelberg University · Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University · Heidelberg Institute for Theoretical Studies (HITS), Heidelberg University +2
PACE-FNO: Physics-Aligned Canonical Equivariance for Fourier Neural Operators
May 18, 2026Jiaxiao Xu, Changhong Mou, Yeyu Zhang +1Fourier Neural Operator OOD Generalization
Shanghai University of Finance and Economics · Utah State University · University of Edinburgh
Symplectic Neural Operators for Learning Infinite Dimensional Hamiltonian Systems
May 15, 2026Yeang Makara, Yusuke Tanaka, Takashi Matsubara +1PDE Operator Learning Neural Operators
Graduate School of Science Kobe University 1-1 Rokkodai-cho, Nada-ku, Kobe, Japan · NTT Communication Science Laboratories 2-4, Hikaridai, Seika-cho, Soraku-gun, Kyoto, Japan · Faculty of Information Science and Technology Hokkaido University Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido, Japan +1
AOT-POT: Adaptive Operator Transformation for Large-Scale PDE Pre-training
May 15, 2026Qitan Lv, Hong Wang, Zhongkai Hao +5Transformer PDE Operator Learning
University of Science and Technology of China · 2Shanghai AI Laboratory · Department of Electronic Engineering, Tsinghua University, China
Eradicating Negative Transfer in Multi-Physics Foundation Models via Sparse Mixture-of-Experts Routing
May 14, 2026Ellwil Sharma, Arastu SharmaSparse Mixture-of-Experts PDE Operator Learning
Shodh AI
Topology-Preserving Neural Operator Learning via Hodge Decomposition
May 13, 2026Dongzhe Zheng, Tao Zhong, Christine Allen-BlanchetteHodge Decomposition PDE Operator Learning
Princeton University
Frequency Bias and OOD Generalization in Neural Operators under a Variable-Coefficient Wave Equation
May 13, 2026Runlong Xie, An LuoFourier Neural Operator PDE Solving
Independent Researcher · School of Statistics, University of Minnesota, MN, USA.
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
Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
May 12, 2026Takashi Furuya, Ryo Ozawa, Jenn-Nan WangPDE Operator Learning Neural Network Approximation Theory
Doshisha University · RIKEN AIP · Tohoku University +1
Neural Quantum Spectral Operator Learning for Solving Partial Differential Equations
May 12, 2026Chanyoung Kim, Myeonghwan Seong, Yujin Kim +2Hybrid Quantum-Classical ML PDE Operator Learning
KAIST · Department of Statistics and Data Science, Yonsei University · Department of Applied Statistics, Yonsei University +2
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
Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs
May 11, 2026Koichi Taniguchi, Sho SonodaPDE Solving PDE Operator Learning
Department of Mathematical and Systems Engineering, Faculty of Engineering Shizuoka University 3-5-1 Johoku, Chuo-ku, Hamamatsu, 432-8561, Japan · RIKEN AIP / CyberAgent 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan
jNO: A JAX Library for Neural Operator and Foundation Model Training
May 11, 2026Leon Armbruster, Rathan Ramesh, Georg Kruse +1PDE Operator Learning Physics-Informed Neural Operators
Fraunhofer Institute for Integrated Systems and Device Technology IISB, Germany
Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators
May 11, 2026Xiaoxiao Lu, Ye Yuan, Jiahao ShiLatent Dynamics Modeling PDE Surrogate Modeling
School of AIA, Huazhong University of Science and Technology
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
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
M $^3$ : Reframing Training Measures for Discretized Physical Simulations
May 9, 2026Yuan Mei, Xingyu Song, Xiaowen Song +1Neural Surrogate Modeling Adaptive Sampling
1Zhejiang University · *Visiting student at The University of Tokyo from Zhejiang University. · 2The University of Tokyo

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Loss-Conditional PINNs for Parametric PDE Families

Anna Lazareva, Alexander Tarakanov

Faculty of Computer Science HSE University, Moscow, Russia · VK and HSE University Moscow, Russia

PDE Solving PDE Operator Learning Physics-Informed ML

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Physics-informed neural networks (PINNs) approximate solutions of ODEs and PDEs by minimising a weighted combination of residual, boundary, initial, and data losses. Their performance is often dominated by the choice of loss weights: a poor weighting can drive training to a degenerate solution in which one physical constraint is satisfied while another is ignored. Existing methods select or adapt a single good set of weights. We take a different view: instead of tuning one weight vector, we explore the entire weight space during training. We introduce LC-PINN, which adapts the loss-conditional training of Dosovitskiy and Djolonga (2020) to the PDE-residual setting: the conditioning vector (either the loss weights or a scalar physical coefficient) is treated as a network input and sampled from a simple prior at every optimisation step. This turns PINN training into learning a continuous family of solutions indexed by that vector, with no solver-generated paired data. LC-PINN thus lies between classical PINNs and operator learning: it stays fully physics-informed but amortises training over a parametric family. Our contribution is not the loss-conditional construction itself, but its extension to PINNs, the unification of the loss-weight and parametric-coefficient regimes under one architecture (concatenation for loss weights, FiLM for coefficients), and a fixed-quadrature L-BFGS finishing protocol that makes the parametric-coefficient regime trainable. We give a lambda-invariance result for the conditional optimum and study LC-PINN on parametric Helmholtz, Schrodinger, viscous Burgers, and Buckley-Leverett equations. A single LC-PINN matches or improves retrained per-weight PINN baselines while parameterising the full family in one model, at a total cost that amortises favourably against per-instance retraining.

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