mlsanityData to 7 Oct 2026

PINNACLE: PINN Adaptive ColLocation and Experimental points selection

arXiv 2404.07662 ↗First posted 11 Apr 2024cs.LG cs.AI physics.comp-phcited by 31

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Abstract (arXiv)

Physics-Informed Neural Networks (PINNs), which incorporate PDEs as soft constraints, train with a composite loss function that contains multiple training point types: different types of collocation points chosen during training to enforce each PDE and initial/boundary conditions, and experimental points which are usually costly to obtain via experiments or simulations. Training PINNs using this loss function is challenging as it typically requires selecting large numbers of points of different types, each with different training dynamics. Unlike past works that focused on the selection of either collocation or experimental points, this work introduces PINN Adaptive ColLocation and Experimental points selection (PINNACLE), the first algorithm that jointly optimizes the selection of all training point types, while automatically adjusting the proportion of collocation point types as training progresses. PINNACLE uses information on the interaction among training point types, which had not been considered before, based on an analysis of PINN training dynamics via the Neural Tangent Kernel (NTK). We theoretically show that the criterion used by PINNACLE is related to the PINN generalization error, and empirically demonstrate that PINNACLE is able to outperform existing point selection methods for forward, inverse, and transfer learning problems.

Key facts. Cited by 31 papers in the map (data through 7 Oct 2026). Builds on 42 of its 68 references that are in the map. First posted on arXiv 11 Apr 2024.

Builds on

42 of 68 references.

References · 42

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  2. 2017Deep Bayesian Active Learning with Image Datacited by 812
  3. 2019Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descentcited by 732
  4. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  5. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  6. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  7. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  8. 2019Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Boundscited by 602
  9. 2019On Exact Computation with an Infinitely Wide Neural Netcited by 572
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  11. 2020A Survey of Deep Active Learningcited by 547
  12. 2019BatchBALD: Efficient and Diverse Batch Acquisition for Deep Bayesian Active Learningcited by 428
30 more
  1. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 396
  2. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  3. 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 233
  4. 2019Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural Networkscited by 221
  5. 2020Physics-informed learning of governing equations from scarce datacited by 167
  6. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  7. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  8. 2021Efficient training of physics-informed neural networks via importance samplingcited by 109
  9. 2021CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Methodcited by 93
  10. 2020Bayesian Deep Ensembles via the Neural Tangent Kernelcited by 90
  11. 2013Revisiting the Nystrom Method for Improved Large-Scale Machine Learningcited by 76
  12. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 75
  13. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
  14. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  15. 2023Separable Physics-Informed Neural Networkscited by 64
  16. 2021Deep Active Learning by Leveraging Training Dynamicscited by 32
  17. 2021NASI: Label- and Data-agnostic Neural Architecture Search at Initializationcited by 28
  18. 2022Making Look-Ahead Active Learning Strategies Feasible with Neural Tangent Kernelscited by 25
  19. 2022Sample-Then-Optimize Batch Neural Thompson Samplingcited by 25
  20. 2022On the Role of Fixed Points of Dynamical Systems in Training Physics-Informed Neural Networkscited by 23
  21. 2021Neural Active Learning with Performance Guaranteescited by 22
  22. 2021Uniform Generalization Bounds for Overparameterized Neural Networkscited by 21
  23. 2022EINNs: Epidemiologically-informed Neural Networkscited by 21
  24. 2022Unifying and Boosting Gradient-Based Training-Free Neural Architecture Searchcited by 17
  25. 2022Federated Neural Banditscited by 17
  26. 2017Scalable Kernel K-Means Clustering with Nystrom Approximation: Relative-Error Boundscited by 16
  27. 2022RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networkscited by 14
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  29. 2023Training-Free Neural Active Learning with Initialization-Robustness Guaranteescited by 5
  30. 2023Does Deep Active Learning Work in the Wild?cited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 6; 2025: 17; 2026: 8.

31 papers.

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  1. 2024Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problemscited by 35
  2. 2024Parameterized Physics-informed Neural Networks for Parameterized PDEscited by 35
  3. 2024Helpful or Harmful Data? Fine-tuning-free Shapley Attribution for Explaining Language Model Predictionscited by 13
  4. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 10
  5. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  6. 2024Investigating Guiding Information for Adaptive Collocation Point Sampling in PINNscited by 3
  7. 2025Neuro-Spectral Architectures for Causal Physics-Informed Networkscited by 3
  8. 2025Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Methodcited by 2
  9. 2025PIED: Physics-Informed Experimental Design for Inverse Problemscited by 2
  10. 2025S2GPT-PINNs: Sparse and Small models for PDEscited by 2
  11. 2025Leveraging Influence Functions for Resampling Data in Physics-Informed Neural Networkscited by 2
  12. 2025Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Featurescited by 2
19 more
  1. 2025Training Deep Physics-Informed Kolmogorov-Arnold Networkscited by 2
  2. 2024PINNfluence: Interpreting PINNs through Influence Functionscited by 1
  3. 2024MelissaDL x Breed: Towards Data-Efficient On-line Supervised Training of Multi-parametric Surrogates with Active Learningcited by 1
  4. 2025Unveiling the Potential of Superexpressive Networks in Implicit Neural Representationscited by 1
  5. 2025A Differentiable Framework for Full and Phaseless Data Inversion Using Neural Implicit Contrast-Source Representationcited by 1
  6. 2026Learning Where to Simulate: Generative Active Sampling for Online PDE Surrogate Trainingcited by 1
  7. 2025Conformal mapping based Physics-informed neural networks for designing neutral inclusionscited by 0
  8. 2025Learning and Transferring Physical Models through Derivativescited by 0
  9. 2025A Cycle-Consistency Constrained Framework for Dynamic Solution Space Reduction in Noninjective Regressioncited by 0
  10. 2025Quasi-Random Physics-informed Neural Networkscited by 0
  11. 2025Uncovering Scaling Laws for Large Language Models via Inverse Problemscited by 0
  12. 2025LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetriescited by 0
  13. 2025More Consistent Accuracy PINN via Alternating Easy-Hard Trainingcited by 0
  14. 2026Diversity-Aware Adaptive Collocation for Physics-Informed Neural Networks via Sparse QUBO Optimization and Hybrid Coresetscited by 0
  15. 2026Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistencycited by 0
  16. 2026Mitigating Gradient Pathology in PINNs through Aligned Constraintcited by 0
  17. 2026PINNs Failure Modes are Overfittingcited by 0
  18. 2026LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networkscited by 0
  19. 2026Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solverscited by 0