# Linearly Constrained Neural Networks

arXiv: https://arxiv.org/abs/2002.01600 · HTML: https://mlsanity.com/paper/2002.01600/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2020.

## Built on by (most cited first, 18 of 18)

- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [RAYEN: Imposition of Hard Convex Constraints on Neural Networks](https://mlsanity.com/paper/2307.08336/) (2023)
- [A New Computationally Simple Approach for Implementing Neural Networks with Output Hard Constraints](https://mlsanity.com/paper/2307.10459/) (2023)
- [Learning to Optimize for Mixed-Integer Non-linear Programming with Feasibility Guarantees](https://mlsanity.com/paper/2410.11061/) (2024)
- [Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations](https://mlsanity.com/paper/2206.09571/) (2022)
- [Neural Fields with Hard Constraints of Arbitrary Differential Order](https://mlsanity.com/paper/2306.08943/) (2023)
- [Towards Scalable Physically Consistent Neural Networks: an Application to Data-driven Multi-zone Thermal Building Models](https://mlsanity.com/paper/2212.12380/) (2022)
- [Incorporating Sum Constraints into Multitask Gaussian Processes](https://mlsanity.com/paper/2202.01793/) (2022)
- [Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations](https://mlsanity.com/paper/2411.16663/) (2024)
- [Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networks](https://mlsanity.com/paper/2503.00755/) (2025)
- [Harmonic (Quantum) Neural Networks](https://mlsanity.com/paper/2212.07462/) (2022)
- [Deep Generative Models with Hard Linear Equality Constraints](https://mlsanity.com/paper/2502.05416/) (2025)
- [Neuro-physical dynamic load modeling using differentiable parametric optimization](https://mlsanity.com/paper/2203.10582/) (2022)
- [Zero Coordinate Shift: Whetted Automatic Differentiation for Physics-informed Operator Learning](https://mlsanity.com/paper/2311.00860/) (2023)
- [Scalable physical source-to-field inference with hypernetworks](https://mlsanity.com/paper/2405.05981/) (2024)
- [LaPON: A Lagrange's-mean-value-theorem-inspired operator network for solving PDEs and its application on NSE](https://mlsanity.com/paper/2505.12360/) (2025)
- [Temperature transferable Machine Learned Coarse Grained model for proteins](https://mlsanity.com/paper/2606.14111/) (2026)
- [Steering Neural Network Training through Interpretable Constraints Based on Partial Dependence](https://mlsanity.com/paper/2607.08641/) (2026)

## Abstract (arXiv)

We present a novel approach to modelling and learning vector fields from physical systems using neural networks that explicitly satisfy known linear operator constraints. To achieve this, the target function is modelled as a linear transformation of an underlying potential field, which is in turn modelled by a neural network. This transformation is chosen such that any prediction of the target function is guaranteed to satisfy the constraints. The approach is demonstrated on both simulated and real data examples.