mlsanityData to 14 Oct 2026

PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs

arXiv 2306.08827 ↗First posted 15 Jun 2023cs.LG cs.NA math.NAcited by 87

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Abstract (arXiv)

While significant progress has been made on Physics-Informed Neural Networks (PINNs), a comprehensive comparison of these methods across a wide range of Partial Differential Equations (PDEs) is still lacking. This study introduces PINNacle, a benchmarking tool designed to fill this gap. PINNacle provides a diverse dataset, comprising over 20 distinct PDEs from various domains, including heat conduction, fluid dynamics, biology, and electromagnetics. These PDEs encapsulate key challenges inherent to real-world problems, such as complex geometry, multi-scale phenomena, nonlinearity, and high dimensionality. PINNacle also offers a user-friendly toolbox, incorporating about 10 state-of-the-art PINN methods for systematic evaluation and comparison. We have conducted extensive experiments with these methods, offering insights into their strengths and weaknesses. In addition to providing a standardized means of assessing performance, PINNacle also offers an in-depth analysis to guide future research, particularly in areas such as domain decomposition methods and loss reweighting for handling multi-scale problems and complex geometry. To the best of our knowledge, it is the largest benchmark with a diverse and comprehensive evaluation that will undoubtedly foster further research in PINNs.

Key facts. Cited by 87 papers in the map (data through 14 Oct 2026). Builds on 38 of its 70 references that are in the map. First posted on arXiv 15 Jun 2023.

Builds on

38 of 70 references.

References · 38

  1. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  2. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  3. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  4. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  5. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  7. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 413
  8. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
  9. 2020GEOM: Energy-annotated molecular conformations for property prediction and molecular generationcited by 243
  10. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  11. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
  12. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 222
26 more
  1. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 197
  2. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 174
  3. 2022Respecting causality is all you need for training physics-informed neural networkscited by 156
  4. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 148
  5. 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
  6. 2017Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modelingcited by 131
  7. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  8. 2021Efficient training of physics-informed neural networks via importance samplingcited by 110
  9. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 98
  10. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  11. 2021Multi-Objective Loss Balancing for Physics-Informed Deep Learningcited by 87
  12. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  13. 2021The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?cited by 61
  14. 2020NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation frameworkcited by 58
  15. 2020Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structurescited by 52
  16. 2019D3M: A deep domain decomposition method for partial differential equationscited by 44
  17. 2021Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networkscited by 43
  18. 2019VarNet: Variational Neural Networks for the Solution of Partial Differential Equationscited by 35
  19. 2019Deep Learning of Subsurface Flow via Theory-guided Neural Networkcited by 32
  20. 2021Data vs. Physics: The Apparent Pareto Front of Physics-Informed Neural Networkscited by 27
  21. 2021Which priors matter? Benchmarking models for learning latent dynamicscited by 26
  22. 2021An Extensible Benchmark Suite for Learning to Simulate Physical Systemscited by 24
  23. 2021ClimART: A Benchmark Dataset for Emulating Atmospheric Radiative Transfer in Weather and Climate Modelscited by 21
  24. 2023MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networkscited by 17
  25. 2022State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learningcited by 12
  26. 2021A Large-Scale Benchmark for the Incompressible Navier-Stokes Equationscited by 4

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 16; 2025: 26; 2026: 43.

87 papers.

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  1. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  2. 2024The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learningcited by 145
  3. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
  4. 2023A Mathematical Guide to Operator Learningcited by 76
  5. 2024ConFIG: Towards Conflict-free Training of Physics Informed Neural Networkscited by 46
  6. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 36
  7. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 28
  8. 2024RoPINN: Region Optimized Physics-Informed Neural Networkscited by 26
  9. 2024Active Learning for Neural PDE Solverscited by 22
  10. 2024The Challenges of the Nonlinear Regime for Physics-Informed Neural Networkscited by 21
  11. 2024Dual Cone Gradient Descent for Training Physics-Informed Neural Networkscited by 21
  12. 2024Preconditioning for Physics-Informed Neural Networkscited by 13
38 more
  1. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 12
  2. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 11
  3. 2025Lang-PINN: From Language to Physics-Informed Neural Networks via a Multi-Agent Frameworkcited by 11
  4. 2025Sub-Sequential Physics-Informed Learning with State Space Modelcited by 10
  5. 2024Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneitycited by 8
  6. 2025Enhancing Physics-Informed Neural Networks Through Feature Engineeringcited by 8
  7. 2025VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Modelscited by 8
  8. 2025HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutionscited by 8
  9. 2026PDEAgent-Bench: A Multi-Metric, Multi-Library Benchmark for PDE Solver Generationcited by 8
  10. 2025Accelerating scientific discovery with the common task frameworkcited by 7
  11. 2025PDE-Agent: A toolchain-augmented multi-agent framework for PDE solvingcited by 7
  12. 2025Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspectivecited by 6
  13. 2025A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learningcited by 6
  14. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  15. 2024Conformalized Prediction of Post-Fault Voltage Trajectories Using Pre-trained and Finetuned Attention-Driven Neural Operatorscited by 5
  16. 2025Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Modelscited by 5
  17. 2025FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulationcited by 5
  18. 2024PAPM: A Physics-aware Proxy Model for Process Systemscited by 4
  19. 2025RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equationscited by 4
  20. 2026Emergent Transfer of a Physics Foundation Model from Simulation to Laboratory Turbulencecited by 4
  21. 2025AutoHood3D: A Multi-Modal Benchmark for Automotive Hood Design and Fluid-Structure Interactioncited by 3
  22. 2026A Score Filter Enhanced Data Assimilation Framework for Data-Driven Dynamical Systemscited by 3
  23. 2024Feature Mapping in Physics-Informed Neural Networks (PINNs)cited by 2
  24. 2024Symmetry Breaking in Neural Network Optimization: Insights from Input Dimension Expansioncited by 2
  25. 2025Scalable Back-Propagation-Free Training of Optical Physics-Informed Neural Networkscited by 2
  26. 2025Re4: Scientific Computing Agent with Rewriting, Resolution, Review and Revisioncited by 2
  27. 2025Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learningcited by 2
  28. 2026The Neural Compiler: Program-to-Network Translation for Hybrid Scientific Machine Learningcited by 2
  29. 2026PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problemscited by 2
  30. 2026EvoPINN: Agentic Discovery of Executable Algorithms for Physics-Informed Neural Networkscited by 2
  31. 2024ConDiff: A Challenging Dataset for Neural Solvers of Partial Differential Equationscited by 1
  32. 2025BWLer: Barycentric Weight Layer Elucidates a Precision-Conditioning Tradeoff for PINNscited by 1
  33. 2025Enhancing Stability of Physics-Informed Neural Network Training Through Saddle-Point Reformulationcited by 1
  34. 2025A new initialisation to Control Gradients in Sinusoidal Neural networkcited by 1
  35. 2026Per-Loss Adapters for Gradient Conflict in Physics-Informed Neural Networkscited by 1
  36. 2026Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospectscited by 1
  37. 2026Fourier Feature Pyramids for Physics-Informed Neural Networkscited by 1
  38. 2026Unveiling Multi-regime Patterns in SciML: Distinct Failure Modes and Regime-specific Optimizationcited by 1

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