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CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method

arXiv 2110.15832 ↗First posted 29 Oct 2021cs.LG cs.CE cs.NAcited by 93

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Abstract (arXiv)

In this study, novel physics-informed neural network (PINN) methods for coupling neighboring support points and their derivative terms which are obtained by automatic differentiation (AD), are proposed to allow efficient training with improved accuracy. The computation of differential operators required for PINNs loss evaluation at collocation points are conventionally obtained via AD. Although AD has the advantage of being able to compute the exact gradients at any point, such PINNs can only achieve high accuracies with large numbers of collocation points, otherwise they are prone to optimizing towards unphysical solution. To make PINN training fast, the dual ideas of using numerical differentiation (ND)-inspired method and coupling it with AD are employed to define the loss function. The ND-based formulation for training loss can strongly link neighboring collocation points to enable efficient training in sparse sample regimes, but its accuracy is restricted by the interpolation scheme. The proposed coupled-automatic-numerical differentiation framework, labeled as can-PINN, unifies the advantages of AD and ND, providing more robust and efficient training than AD-based PINNs, while further improving accuracy by up to 1-2 orders of magnitude relative to ND-based PINNs. For a proof-of-concept demonstration of this can-scheme to fluid dynamic problems, two numerical-inspired instantiations of can-PINN schemes for the convection and pressure gradient terms were derived to solve the incompressible Navier-Stokes (N-S) equations. The superior performance of can-PINNs is demonstrated on several challenging problems, including the flow mixing phenomena, lid driven flow in a cavity, and channel flow over a backward facing step. The results reveal that for challenging problems like these, can-PINNs can consistently achieve very good accuracy whereas conventional AD-based PINNs fail.

Key facts. Cited by 93 papers in the map (data through 7 Oct 2026). Builds on 32 of its 74 references that are in the map. First posted on arXiv 29 Oct 2021.

Builds on

32 of 74 references.

References · 32

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  10. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  11. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  12. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 134
20 more
  1. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  2. 2019Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networkscited by 127
  3. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  4. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  5. 2021Efficient training of physics-informed neural networks via importance samplingcited by 109
  6. 2021Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problemscited by 100
  7. 2021PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEscited by 97
  8. 2019Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networkscited by 76
  9. 2018Deep Learning of Vortex Induced Vibrationscited by 74
  10. 2020Physics-informed neural network for ultrasound nondestructive quantification of surface breaking crackscited by 67
  11. 2021Learning in Sinusoidal Spaces with Physics-Informed Neural Networkscited by 61
  12. 2020NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation frameworkcited by 58
  13. 2020A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applicationscited by 53
  14. 2020Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structurescited by 52
  15. 2020Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacturecited by 48
  16. 2020DiscretizationNet: A Machine-Learning based solver for Navier-Stokes Equations using Finite Volume Discretizationcited by 35
  17. 2020Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3Dcited by 26
  18. 2020A nonlocal physics-informed deep learning framework using the peridynamic differential operatorcited by 21
  19. 2021Can Transfer Neuroevolution Tractably Solve Your Differential Equations?cited by 11
  20. 2018Physics-Driven Regularization of Deep Neural Networks for Enhanced Engineering Design and Analysiscited by 6

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 11; 2023: 26; 2024: 23; 2025: 17; 2026: 16.

93 papers.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  2. 2023An Expert's Guide to Training Physics-informed Neural Networkscited by 119
  3. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  4. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 70
  5. 2023Separable Physics-Informed Neural Networkscited by 64
  6. 2025Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspectivecited by 55
  7. 2023Multi-stage Neural Networks: Function Approximator of Machine Precisioncited by 48
  8. 2023CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamicscited by 40
  9. 2024PINNACLE: PINN Adaptive ColLocation and Experimental points selectioncited by 31
  10. 2022Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computingcited by 24
  11. 2022Learning Physics-Informed Neural Networks without Stacked Back-propagationcited by 23
  12. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
38 more
  1. 2022Fully probabilistic deep models for forward and inverse problems in parametric PDEscited by 19
  2. 2023Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEscited by 19
  3. 2025Vid2Sim: Generalizable, Video-based Reconstruction of Appearance, Geometry and Physics for Mesh-free Simulationcited by 18
  4. 2023Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domainscited by 14
  5. 2024Physics Informed Distillation for Diffusion Modelscited by 13
  6. 2023Tensor-Compressed Back-Propagation-Free Training for (Physics-Informed) Neural Networkscited by 12
  7. 2024Data-Driven Physics-Informed Neural Networks: A Digital Twin Perspectivecited by 12
  8. 2023Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrologycited by 11
  9. 2024Functional Tensor Decompositions for Physics-Informed Neural Networkscited by 11
  10. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 10
  11. 2024Precise-Physics Driven Text-to-3D Generationcited by 9
  12. 2024Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equationscited by 9
  13. 2022Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Mediacited by 8
  14. 2023Random Grid Neural Processes for Parametric Partial Differential Equationscited by 8
  15. 2023Physics-constrained robust learning of open-form partial differential equations from limited and noisy datacited by 8
  16. 2024Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problemcited by 8
  17. 2023On the locality of local neural operator in learning fluid dynamicscited by 6
  18. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  19. 2022A Curriculum-Training-Based Strategy for Distributing Collocation Points during Physics-Informed Neural Network Trainingcited by 5
  20. 2023Temporal Difference Learning for High-Dimensional PIDEs with Jumpscited by 5
  21. 2023Evolutionary Optimization of Physics-Informed Neural Networks: Advancing Generalizability by the Baldwin Effectcited by 5
  22. 2024DOF: Accelerating High-order Differential Operators with Forward Propagationcited by 5
  23. 2024Spatio-Temporal Fluid Dynamics Modeling via Physical-Awareness and Parameter Diffusion Guidancecited by 5
  24. 2024Data-Guided Physics-Informed Neural Networks for Solving Inverse Problems in Partial Differential Equationscited by 5
  25. 2022RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Networkcited by 4
  26. 2023LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometrycited by 4
  27. 2023ParticleWNN: a Novel Neural Networks Framework for Solving Partial Differential Equationscited by 4
  28. 2023Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanicscited by 4
  29. 2024Separable Operator Networkscited by 4
  30. 2024Domain-decoupled Physics-informed Neural Networks with Closed-form Gradients for Fast Model Learning of Dynamical Systemscited by 4
  31. 2025NeuSpring: Neural Spring Fields for Reconstruction and Simulation of Deformable Objects from Videoscited by 4
  32. 2026Out-of-Distribution Generalization for Neural Physics Solverscited by 4
  33. 2026Scale-PINN: Learning Efficient Physics-Informed Neural Networks Through Sequential Correctioncited by 4
  34. 2022Robustness of Physics-Informed Neural Networks to Noise in Sensor Datacited by 3
  35. 2023PMNN:Physical Model-driven Neural Network for solving time-fractional differential equationscited by 3
  36. 2024ImplicitTerrain: a Continuous Surface Model for Terrain Data Analysiscited by 3
  37. 2024Solving Differential Equations with Constrained Learningcited by 3
  38. 2025Enforcing hidden physics in physics-informed neural networkscited by 3

50 most cited of 93 All 93 (JSON)