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

arXiv: https://arxiv.org/abs/2109.14860 · HTML: https://mlsanity.com/paper/2109.14860/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 77 references that are in the map.
- First posted on arXiv 30 Sep 2021.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Analysis 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 Equations](https://mlsanity.com/paper/1809.03062/) (2018)
- [Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers](https://mlsanity.com/paper/2008.09768/) (2020)
- [SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs](https://mlsanity.com/paper/2102.13037/) (2021)
- [Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures](https://mlsanity.com/paper/1910.09045/) (2019)
- [Training neural networks under physical constraints using a stochastic augmented Lagrangian approach](https://mlsanity.com/paper/2009.07330/) (2020)
- [Physics-informed machine learning for sensor fault detection with flight test data](https://mlsanity.com/paper/2006.13380/) (2020)

## Built on by (most cited first, 30 of 38)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks](https://mlsanity.com/paper/2407.01613/) (2024)
- [Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)](https://mlsanity.com/paper/2209.09988/) (2022)
- [An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks](https://mlsanity.com/paper/2306.04904/) (2023)
- [Machine learning and domain decomposition methods -- a survey](https://mlsanity.com/paper/2312.14050/) (2023)
- [Learning in PINNs: Phase transition, diffusion equilibrium, and generalization](https://mlsanity.com/paper/2403.18494/) (2024)
- [Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2205.06948/) (2022)
- [Neural topology optimization: the good, the bad, and the ugly](https://mlsanity.com/paper/2407.13954/) (2024)
- [Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations](https://mlsanity.com/paper/2303.11577/) (2023)
- [BiLO: Bilevel Local Operator Learning for PDE Inverse Problems](https://mlsanity.com/paper/2404.17789/) (2024)
- [Multi-Fidelity Methods for Optimization: A Survey](https://mlsanity.com/paper/2402.09638/) (2024)
- [A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning](https://mlsanity.com/paper/2509.14198/) (2025)
- [Self-adaptive weighting and sampling for physics-informed neural networks](https://mlsanity.com/paper/2511.05452/) (2025)
- [Data-Guided Physics-Informed Neural Networks for Solving Inverse Problems in Partial Differential Equations](https://mlsanity.com/paper/2407.10836/) (2024)
- [Residual-Quantile Adjustment for Adaptive Training of Physics-informed Neural Network](https://mlsanity.com/paper/2209.05315/) (2022)
- [Robust Regression with Highly Corrupted Data via Physics Informed Neural Networks](https://mlsanity.com/paper/2210.10646/) (2022)
- [A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networks](https://mlsanity.com/paper/2307.12435/) (2023)
- [Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Data](https://mlsanity.com/paper/2406.02581/) (2024)
- [Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks](https://mlsanity.com/paper/2409.13644/) (2024)
- [Solving Differential Equations with Constrained Learning](https://mlsanity.com/paper/2410.22796/) (2024)
- [Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems](https://mlsanity.com/paper/2508.15695/) (2025)
- [ASPINN: An asymptotic strategy for solving singularly perturbed differential equations](https://mlsanity.com/paper/2409.13185/) (2024)
- [PC-SRGAN: Physically Consistent Super-Resolution Generative Adversarial Network for General Transient Simulations](https://mlsanity.com/paper/2505.06502/) (2025)
- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)
- [MRF-PINN: A Multi-Receptive-Field convolutional physics-informed neural network for solving partial differential equations](https://mlsanity.com/paper/2209.03151/) (2022)
- [Improved physics-informed neural network in mitigating gradient related failures](https://mlsanity.com/paper/2407.19421/) (2024)
- [Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks](https://mlsanity.com/paper/2503.20222/) (2025)
- [Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method](https://mlsanity.com/paper/2505.04875/) (2025)

## 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.