# Physics-Informed Neural Networks with Hard Linear Equality Constraints

arXiv: https://arxiv.org/abs/2402.07251 · HTML: https://mlsanity.com/paper/2402.07251/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 62 references that are in the map.
- First posted on arXiv 11 Feb 2024.

## Builds on (most cited first, 19 of 19)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Differentiable Convex Optimization Layers](https://mlsanity.com/paper/1910.12430/) (2019)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Partition-based formulations for mixed-integer optimization of trained ReLU neural networks](https://mlsanity.com/paper/2102.04373/) (2021)
- [OMLT: Optimization & Machine Learning Toolkit](https://mlsanity.com/paper/2202.02414/) (2022)
- [Global Deterministic Optimization with Artificial Neural Networks Embedded](https://mlsanity.com/paper/1801.07114/) (2018)
- [ReLU Networks as Surrogate Models in Mixed-Integer Linear Programs](https://mlsanity.com/paper/1907.03140/) (2019)
- [Training neural networks under physical constraints using a stochastic augmented Lagrangian approach](https://mlsanity.com/paper/2009.07330/) (2020)
- [Stochastic Physics-Informed Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.01621/) (2021)
- [The ALAMO approach to machine learning](https://mlsanity.com/paper/1705.10918/) (2017)
- [Global Optimization of Gaussian processes](https://mlsanity.com/paper/2005.10902/) (2020)
- [A Physics-informed Deep Learning Approach for Minimum Effort Stochastic Control of Colloidal Self-Assembly](https://mlsanity.com/paper/2208.09182/) (2022)
- [A robust approach to warped Gaussian process-constrained optimization](https://mlsanity.com/paper/2006.08222/) (2020)

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

- [HardNet: Hard-Constrained Neural Networks with Universal Approximation Guarantees](https://mlsanity.com/paper/2410.10807/) (2024)
- [ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection](https://mlsanity.com/paper/2502.06774/) (2025)
- [Physics-Informed Neural Networks with Hard Nonlinear Equality and Inequality Constraints](https://mlsanity.com/paper/2507.08124/) (2025)
- [Enforcing Hard Linear Constraints in Deep Learning Models with Decision Rules](https://mlsanity.com/paper/2505.13858/) (2025)
- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [Physics-Constrained Machine Learning for Chemical Engineering](https://mlsanity.com/paper/2508.20649/) (2025)
- [Picard-KKT-hPINN: Enforcing Nonlinear Enthalpy Balances for Physically Consistent Neural Networks](https://mlsanity.com/paper/2501.17782/) (2025)
- [Federated Learning from Molecules to Processes: A Perspective](https://mlsanity.com/paper/2506.18525/) (2025)
- [DAE-HardNet: A Physics Constrained Neural Network Enforcing Differential-Algebraic Hard Constraints](https://mlsanity.com/paper/2512.05881/) (2025)
- [ECO: Energy-Constrained Operator Learning for Chaotic Dynamics with Boundedness Guarantees](https://mlsanity.com/paper/2512.01984/) (2025)
- [SnareNet: Flexible Repair Layers for Neural Networks with Hard Constraints](https://mlsanity.com/paper/2602.09317/) (2026)
- [HardNet++: Nonlinear Constraint Enforcement in Neural Networks](https://mlsanity.com/paper/2604.19669/) (2026)
- [Deep Generative Models with Hard Linear Equality Constraints](https://mlsanity.com/paper/2502.05416/) (2025)
- [Machine Learning for Physical Simulation Challenge Results and Retrospective Analysis: Power Grid Use Case](https://mlsanity.com/paper/2505.01156/) (2025)
- [Learning with Embedded Linear Equality Constraints via Variational Bayesian Inference](https://mlsanity.com/paper/2604.24911/) (2026)
- [AdamFLIP: Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN Training](https://mlsanity.com/paper/2605.08408/) (2026)
- [Exact Fixed-Point Constraints in Neural-ODEs with Provable Universality](https://mlsanity.com/paper/2605.10613/) (2026)
- [A condensing approach to multiple shooting neural ordinary differential equation](https://mlsanity.com/paper/2506.00724/) (2025)
- [A Neural Network Framework for Discovering Closed-form Solutions to Quadratic Programs with Linear Constraints](https://mlsanity.com/paper/2510.23737/) (2025)
- [Guaranteeing Conservation of Integrals with Projection in Physics-Informed Neural Networks](https://mlsanity.com/paper/2511.09048/) (2025)
- [Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNs](https://mlsanity.com/paper/2603.29237/) (2026)
- [NLPOpt-Net: A Learning Method for Nonlinear Optimization with Feasibility Guarantees](https://mlsanity.com/paper/2605.00260/) (2026)
- [CAffNet: Hard Constraint-Affine Neural Networks](https://mlsanity.com/paper/2605.24437/) (2026)
- [DisjunctiveNet: Neural Symbolic Learning via Differentiable Convexified Optimization Layers](https://mlsanity.com/paper/2605.30456/) (2026)
- [PL-KKT-hPINN: Enforcing Nonlinear Equality Constraints on Neural Networks via Piecewise-Linear Projection](https://mlsanity.com/paper/2606.10682/) (2026)
- [Structure-Preserving Neural Surrogates with Tractable Uncertainty Quantification](https://mlsanity.com/paper/2606.11650/) (2026)
- [Embedding Linear Equality Constraints in Probabilistic Neural Networks for Dynamic Modelling](https://mlsanity.com/paper/2606.21728/) (2026)
- [An Embedded RISC-V Evaluation of Kolmogorov--Arnold Networks in Hard-Constrained Recurrent Physics-Informed Models](https://mlsanity.com/paper/2608.00737/) (2026)
- [SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization](https://mlsanity.com/paper/2609.33515/) (2026)
- [Atoms to Processes: The Role of Artificial Intelligence and Machine Learning in Chemical Engineering](https://mlsanity.com/paper/2610.02014/) (2026)

## Abstract (arXiv)

Surrogate modeling is used to replace computationally expensive simulations. Neural networks have been widely applied as surrogate models that enable efficient evaluations over complex physical systems. Despite this, neural networks are data-driven models and devoid of any physics. The incorporation of physics into neural networks can improve generalization and data efficiency. The physics-informed neural network (PINN) is an approach to leverage known physical constraints present in the data, but it cannot strictly satisfy them in the predictions. This work proposes a novel physics-informed neural network, KKT-hPINN, which rigorously guarantees hard linear equality constraints through projection layers derived from KKT conditions. Numerical experiments on Aspen models of a continuous stirred-tank reactor (CSTR) unit, an extractive distillation subsystem, and a chemical plant demonstrate that this model can further enhance the prediction accuracy.