PDE Surrogate Modeling

PDE: Partial Differential Equation

PDE surrogate modeling studies how to approximate the mapping from physical conditions, geometry, and parameters to a PDE’s solution, reducing the cost of repeated simulations. Methods range from reduced-order models such as POD and latent dynamics to neural operators such as Fourier Neural Operators and DeepONets, as well as mesh-based graph networks, transformers, and generative models. Many approaches combine these learned models with physical structure, including conservation laws, symmetries, boundary conditions, and numerical solvers, to improve reliability and support inverse problems or design optimization. A central challenge is balancing speed with accuracy under new geometries or regimes, stable long rollouts, fine-scale fidelity, and calibrated uncertainty; whether a surrogate saves time overall depends on training and deployment costs as well as inference speed.

Latest papers

331 papers50 papers in the last four weeks, up 72% on the four weeks before. 0.3% of all new papers.

Breakeven complexity: A new perspective on neural partial differential equation solvers
May 14, 2026Yijing Zhang, Nicholas Roberts, Tanya Marwah +1Neural PDE Solvers Benchmark Design
University of Wisconsin–Madison · Google DeepMind
Tadpole: Autoencoders as Foundation Models for 3D PDEs with Online Learning
May 14, 2026Qiang Liu, Felix Koehler, Benjamin Holzschuh +13D Foundation Models PDE Surrogate Modeling
TUM School of Computation, Information and Technology, Technical University of Munich, Garching, Germany · MCML, Munich Center for Machine Learning, Munich, Germany
Discovering Physical Directions in Weight Space: Composing Neural PDE Experts
May 14, 2026Pengkai Wang, Pengwei Liu, Yuanyi Wang +7PDE Surrogate Modeling Model Merging
1The Hong Kong Polytechnic University · 2Zhejiang University · 3Tsinghua University +1
Toward AI-Driven Digital Twins for Metropolitan Floods: A Conditional Latent Dynamics Network Surrogate of the Shallow Water Equations
May 13, 2026Phillip Si, Yuan Qiu, Omar Sallam +4Digital Twins PDE Surrogate Modeling
School of Computational Science and Engineering, Georgia Institute of Technology, 756 West Peachtree Street Northwest, Atlanta, 30308, GA, USA · Environmental Science Division, Argonne National Laboratory, 9700 S Cass Ave, Lemont, 60439, IL, USA
U-HNO: A U-shaped Hybrid Neural Operator with Sparse-Point Adaptive Routing for Non-stationary PDE Dynamics
May 13, 2026Yingzhe Ma, Xiao Yang, Yuxin Xie +2Fourier Neural Operator PDE Surrogate Modeling
University of Electronic Science and Technology of China · Peking University
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
Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance
May 9, 2026Xiaofeng LiuPDE Solving PDE Surrogate Modeling
Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, PA, USA · Institute for Computational and Data Sciences, Pennsylvania State University, University Park, PA, USA
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
PACT: Peak-Aware Cross-Attention Graph Transformers for Efficient Storm-Surge Emulation
May 9, 2026Zesheng Liu, Doyup Kwon, Ning Lin +1PDE Surrogate Modeling Spatiotemporal Forecasting
Department of Computer Science and Engineering, Lehigh University · Department of Civil and Environmental Engineering, Princeton University · Department of Civil and Environmental Engineering, Lehigh University
Can We Formally Verify Neural PDE Surrogates? SMT Compilation of Small Fourier Neural Operators
May 9, 2026Ali Baheri, Ignacio Laguna PeraltaNeural Network Verification Fourier Neural Operator
Rochester Institute of Technology, Rochester, NY, USA · Lawrence Livermore National Laboratory, Livermore, CA, USA
Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow
May 9, 2026Hanru Bai, Yuncheng Zhou, Difan ZouFlow Matching Neural PDE Solvers
1Fudan University · 2The University of Hong Kong.
A meshfree exterior calculus for generalizable and data-efficient learning of physics from point clouds
May 8, 2026Benjamin D. Shaffer, Brooks Kinch, M. Ani Hsieh +1Point Cloud Learning PDE Surrogate Modeling
University of Pennsylvania Philadelphia, PA 19104 · *Mechanical Engineering and Applied Mechanics
Accelerated and data-efficient flow prediction in stirred tanks via physics-informed learning
May 8, 2026Mahdi Naderibeni, Liang Wu, David M. J. TaxPDE Surrogate Modeling Physics-Informed ML
aPattern Recognition and Bio-informatics Group, Delft University of Technology, Delft, the Netherlands · bScience & Research and Innovation, dsm-firmenich, Delft, the Netherlands
Learned Lagrangian Models of PDEs via Euler-Lagrange Residual Minimization
May 8, 2026Lyra Zhornyak, Eric Forgoston, M. Ani HsiehNeural PDE Solvers PDE Surrogate Modeling
University of Pennsylvania · Montclair State University
AeroJEPA: Learning Semantic Latent Representations for Scalable 3D Aerodynamic Field Modeling
May 7, 2026Francisco Giral, Abhijeet Vishwasrao, Andrea Arroyo Ramo +8Joint-Embedding Predictive Architecture PDE Surrogate Modeling
Universidad Politécnica de Madrid · University of Michigan · Universitat Politècnica de València +3
Towards Scalable One-Step Generative Modeling for Autoregressive Dynamical System Forecasting
May 7, 2026Tianyue Yang, Xiao XuePDE Surrogate Modeling One-Step Generative Modeling
The Center for Computational Science University College London
Posterior-First Neural PDE Simulation: Inferring What Downstream Tasks Need from a Single Field
May 5, 2026Wenshuo Wang, Fan ZhangNeural PDE Solvers PDE Surrogate Modeling
School of Future Technology, South China University of Technology, China · State Key Laboratory of Ocean Sensing & Ocean College, Zhejiang University, China · Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, USA
Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogates
May 4, 2026Xinyu Li, Jianhua Zhang, Liang ChenMultiphysics Simulation PDE Surrogate Modeling
School of Microelectronics, Shanghai University, Shanghai 201800, China
Mesh Based Simulations with Spatial and Temporal awareness
May 2, 2026Paul Garnier, Vincent Lannelongue, Elie HachemGraph Neural Networks Neural PDE Solvers
1CEMEF - Mines Paris PSL
Adaptation of AI-accelerated CFD Simulations to the IPU platform
May 1, 2026P. Rosciszewski, A. Krzywaniak, S. Iserte +2AI for Science PDE Surrogate Modeling
Graphcore Poland · Dpt. of Construction and Mechanical Engineering, Universitat Jaume I, Spain · Barcelona Supercomputing Center, Spain +2
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
Acceleration of horizontal numerical advection for atmospheric modeling through surrogate modeling with temporal coarse-graining
Apr 30, 2026Manho Park, Christopher V. Rackauckas, Christopher W. TessumPDE Surrogate Modeling Computational Fluid Dynamics
The Grainger College of Engineering, Department of Civil and Environmental Engineering, University of Illinois Urbana-Champaign · Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, USA
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
Multi-scale Dynamic Wake Modeling and Prediction of Floating Offshore Wind Turbines via Physics-Informed Neural Networks and Fourier Neural Operators
Apr 27, 2026Guodan Dong, Jianhua Qin, Chang XuFourier Neural Operator PDE Surrogate Modeling
College of Renewable Energy Hohai University Changzhou 213200 China · College of Water Conservancy and Hydropower Engineering Nanjing 210098 China · National Technology Innovation Center for Wind Power Hohai University Changzhou 213200 China
GeoFunFlow-3D: A Physics-Guided Generative Flow Matching Framework for High-Fidelity 3D Aerodynamic Inference over Complex Geometries
Apr 25, 2026Ruiling Jiang, Yong Zhang, Houbiao LiPDE Surrogate Modeling Generative Modeling
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China
Neural surrogates for crystal growth dynamics with variable supersaturation: explicit vs. implicit conditioning
Apr 23, 2026Matteo Rigoni, Daniele Lanzoni, Francesco Montalenti +1Neural Surrogate Modeling Convolutional RNNs
Dept. of Materials Science, University of Milano-Bicocca, 20125 Milano, Italy · Dept. of Physics, University of Genova, 16146 Genova, Italy
A Deep U-Net Framework for Flood Hazard Mapping Using Hydraulic Simulations of the Wupper Catchment
Apr 22, 2026Christian Lammers, Fernando Arévalo, Leonie Märker-Neuhaus +3Neural Surrogate Modeling U-Net
Institute for Computing and Information Sciences Radboud University Nijmegen, Netherlands · Wupper Association (Wupperverband) Wuppertal, Germany
FlowForge: A Staged Local Rollout Engine for Flow-Field Prediction
Apr 21, 2026Xiaowen Zhang, Ziming Zhou, Fengnian Zhao +1Efficient Inference PDE Surrogate Modeling
Shanghai Jiao Tong University · University of Michigan
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

May 14, 2026J/K move · Enter open · S save

Breakeven complexity: A new perspective on neural partial differential equation solvers

Yijing Zhang, Nicholas Roberts, Tanya Marwah, Mikhail Khodak

University of Wisconsin–Madison · Google DeepMind

Neural PDE Solvers Benchmark Design PDE Surrogate Modeling

Open paper AskPDF arXiv

Neural surrogate solvers of partial differential equations (PDEs) promise dramatic speedups over numerical methods, especially in scenarios requiring many solves. However, current accuracy-based evaluations do not fully consider two central issues: (1) neural solvers incur substantial up-front costs for data generation, training, and tuning; and (2) classical solvers can also generate low-fidelity solutions at a sufficiently low simulation cost. To explicitly account for these realities and fully incorporate end-to-end costs, we propose an evaluation framework centered on breakeven complexity, a metric that counts the forward solves before a learned solver is cost-effective relative to an error-equivalent traditional solver. To evaluate this measure, we apply scaling laws to determine how much training budget to allocate to data generation and discuss how to achieve smooth error-matching in diverse settings. We evaluate the breakeven complexity of multiple neural PDE solvers on three PDEs on 2D periodic domains from APEBench and a novel benchmark of flows past multiple obstacles generated by the GPU-native PyFR code. Among other findings, our results suggest that neural PDE solvers become more effective as problems get harder in terms of cost, dimension, rollout, physics regime (e.g. higher Reynolds number), etc.