mlsanityData to 7 Oct 2026

Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion

arXiv 2109.14860 ↗First posted 30 Sep 2021physics.comp-ph cs.LG cs.NAcited by 38

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Abstract (arXiv)

Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE). In PINNs, the residual form of the PDE of interest and its boundary conditions are lumped into a composite objective function as soft penalties. Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach when applied to different kinds of PDEs. To address these limitations, we propose a versatile framework based on a constrained optimization problem formulation, where we use the augmented Lagrangian method (ALM) to constrain the solution of a PDE with its boundary conditions and any high-fidelity data that may be available. Our approach is adept at forward and inverse problems with multi-fidelity data fusion. We demonstrate the efficacy and versatility of our physics- and equality-constrained deep-learning framework by applying it to several forward and inverse problems involving multi-dimensional PDEs. Our framework achieves orders of magnitude improvements in accuracy levels in comparison with state-of-the-art physics-informed neural networks.

Key facts. Cited by 38 papers in the map (data through 7 Oct 2026). Builds on 24 of its 77 references that are in the map. First posted on arXiv 30 Sep 2021.

Builds on

24 of 77 references.

References · 24

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  3. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  4. 2014Neural Machine Translation by Jointly Learning to Align and Translatecited by 4,530
  5. 2014Sequence to Sequence Learning with Neural Networkscited by 2,974
  6. 2016Identity Mappings in Deep Residual Networkscited by 1,960
  7. 2014Deep Learning in Neural Networks: An Overviewcited by 765
  8. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  9. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  10. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  11. 2019Machine Learning for Fluid Mechanicscited by 380
  12. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
12 more
  1. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  2. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  3. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  4. 2019Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networkscited by 76
  5. 2018Deep Learning of Vortex Induced Vibrationscited by 74
  6. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  7. 2018Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equationscited by 44
  8. 2020Hierarchical Deep Learning of Multiscale Differential Equation Time-Stepperscited by 37
  9. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  10. 2019Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architecturescited by 21
  11. 2020Training neural networks under physical constraints using a stochastic augmented Lagrangian approachcited by 19
  12. 2020Physics-informed machine learning for sensor fault detection with flight test datacited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 7; 2023: 6; 2024: 13; 2025: 11; 2026: 1.

38 papers.

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  1. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  2. 2023Residual-based attention in physics-informed neural networkscited by 72
  3. 2022Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)cited by 35
  4. 2024Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networkscited by 20
  5. 2022Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)cited by 17
  6. 2023An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networkscited by 15
  7. 2023Machine learning and domain decomposition methods -- a surveycited by 14
  8. 2024Learning in PINNs: Phase transition, diffusion equilibrium, and generalizationcited by 13
  9. 2022Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Datacited by 12
  10. 2024Neural topology optimization: the good, the bad, and the uglycited by 9
  11. 2023Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equationscited by 8
  12. 2024BiLO: Bilevel Local Operator Learning for PDE Inverse Problemscited by 8
26 more
  1. 2025A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learningcited by 6
  2. 2025Self-adaptive weighting and sampling for physics-informed neural networkscited by 6
  3. 2024Multi-Fidelity Methods for Optimization: A Surveycited by 5
  4. 2024Data-Guided Physics-Informed Neural Networks for Solving Inverse Problems in Partial Differential Equationscited by 5
  5. 2022Residual-Quantile Adjustment for Adaptive Training of Physics-informed Neural Networkcited by 4
  6. 2022Robust Regression with Highly Corrupted Data via Physics Informed Neural Networkscited by 4
  7. 2023A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networkscited by 4
  8. 2024Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Datacited by 3
  9. 2024Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networkscited by 3
  10. 2024Solving Differential Equations with Constrained Learningcited by 3
  11. 2025Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problemscited by 3
  12. 2024ASPINN: An asymptotic strategy for solving singularly perturbed differential equationscited by 2
  13. 2025PC-SRGAN: Physically Consistent Super-Resolution Generative Adversarial Network for General Transient Simulationscited by 2
  14. 2022MRF-PINN: A Multi-Receptive-Field convolutional physics-informed neural network for solving partial differential equationscited by 1
  15. 2024Improved physics-informed neural network in mitigating gradient related failurescited by 1
  16. 2025Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networkscited by 1
  17. 2025Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force methodcited by 1
  18. 2025Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometriescited by 1
  19. 2022Characterizing and Mitigating the Difficulty in Training Physics-informed Artificial Neural Networks under Pointwise Constraintscited by 0
  20. 2023The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approachcited by 0
  21. 2024Equation identification for fluid flows via physics-informed neural networkscited by 0
  22. 2025Prediction of acoustic field in 1-D uniform duct with varying mean flow and temperature using neural networkscited by 0
  23. 2025NASP-T: A Fuzzy Neuro-Symbolic Transformer for Logic-Constrained Aviation Safety Report Classificationcited by 0
  24. 2025Unsupervised simulation of incompressible flows with physics- and equality- constrained artificial neural networkscited by 0
  25. 2025Fundamentals of Regressioncited by 0
  26. 2026Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNscited by 0