# Hard Constraint Projection in a Physics Informed Neural Network

arXiv: https://arxiv.org/abs/2601.06244 · HTML: https://mlsanity.com/paper/2601.06244/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 8 references that are in the map.
- First posted on arXiv 9 Jan 2026.

## Builds on (most cited first, 5 of 5)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Theory-guided hard constraint projection (HCP): a knowledge-based data-driven scientific machine learning method](https://mlsanity.com/paper/2012.06148/) (2020)

## Built on by (most cited first, 2 of 2)

- [Stochastic Dimension Implicit Functional Projections for Global Integral Conservation in High-Dimensional PINNs](https://mlsanity.com/paper/2603.29237/) (2026)
- [SLP-ProbHard: Probabilistic Hard-Constrained Learning via Structural Latent Parameterization](https://mlsanity.com/paper/2609.33515/) (2026)

## Abstract (arXiv)

In this work, we embed hard constraints in a physics informed neural network (PINN) which predicts solutions to the 2D incompressible Navier Stokes equations. We extend the hard constraint method introduced by Chen et al. (arXiv:2012.06148) from a linear PDE to a strongly non-linear PDE. The PINN is used to estimate the stream function and pressure of the fluid, and by differentiating the stream function we can recover an incompressible velocity field. An unlearnable hard constraint projection (HCP) layer projects the predicted velocity and pressure to a hyperplane that admits only exact solutions to a discretised form of the governing equations.