# Picard-KKT-hPINN: Enforcing Nonlinear Enthalpy Balances for Physically Consistent Neural Networks

arXiv: https://arxiv.org/abs/2501.17782 · HTML: https://mlsanity.com/paper/2501.17782/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 8 references that are in the map.
- First posted on arXiv 29 Jan 2025.

## Builds on (most cited first, 3 of 3)

- [Physics-Informed Neural Networks with Hard Linear Equality Constraints](https://mlsanity.com/paper/2402.07251/) (2024)
- [Theory-guided hard constraint projection (HCP): a knowledge-based data-driven scientific machine learning method](https://mlsanity.com/paper/2012.06148/) (2020)
- [Obey validity limits of data-driven models](https://mlsanity.com/paper/2010.03405/) (2020)

## Built on by (most cited first, 3 of 3)

- [ENFORCE: Nonlinear Constrained Learning with Adaptive-depth Neural Projection](https://mlsanity.com/paper/2502.06774/) (2025)
- [Physics-Constrained Machine Learning for Chemical Engineering](https://mlsanity.com/paper/2508.20649/) (2025)
- [PL-KKT-hPINN: Enforcing Nonlinear Equality Constraints on Neural Networks via Piecewise-Linear Projection](https://mlsanity.com/paper/2606.10682/) (2026)

## Abstract (arXiv)

Neural networks are widely used as surrogate models but they do not guarantee physically consistent predictions thereby preventing adoption in various applications. We propose a method that can enforce NNs to satisfy physical laws that are nonlinear in nature such as enthalpy balances. Our approach, inspired by Picard successive approximations method, aims to enforce multiplicatively separable constraints by sequentially freezing and projecting a set of the participating variables. We demonstrate our PicardKKThPINN for surrogate modeling of a catalytic packed bed reactor for methanol synthesis. Our results show that the method efficiently enforces nonlinear enthalpy and linear atomic balances at machine-level precision. Additionally, we show that enforcing conservation laws can improve accuracy in data-scarce conditions compared to vanilla multilayer perceptron.