# Projected Neural Differential Equations for Learning Constrained Dynamics

arXiv: https://arxiv.org/abs/2410.23667 · HTML: https://mlsanity.com/paper/2410.23667/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 56 of its 110 references that are in the map.
- First posted on arXiv 31 Oct 2024.

## Builds on (most cited first, 30 of 56)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Neural General Circulation Models for Weather and Climate](https://mlsanity.com/paper/2311.07222/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Latent Variable Modelling with Hyperbolic Normalizing Flows](https://mlsanity.com/paper/2002.06336/) (2020)
- [LEADS: Learning Dynamical Systems that Generalize Across Environments](https://mlsanity.com/paper/2106.04546/) (2021)
- [Adaptive-Control-Oriented Meta-Learning for Nonlinear Systems](https://mlsanity.com/paper/2103.04490/) (2021)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)
- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [Hard-Constrained Deep Learning for Climate Downscaling](https://mlsanity.com/paper/2208.05424/) (2022)

## Built on by (most cited first, 5 of 5)

- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [Causal Climate Emulation with Bayesian Filtering](https://mlsanity.com/paper/2506.09891/) (2025)
- [A Simultaneous Approach for Training Neural Differential-Algebraic Systems of Equations](https://mlsanity.com/paper/2504.04665/) (2025)
- [ArchesClimate: Probabilistic Decadal Ensemble Generation With Flow Matching](https://mlsanity.com/paper/2509.15942/) (2025)
- [Exact Fixed-Point Constraints in Neural-ODEs with Provable Universality](https://mlsanity.com/paper/2605.10613/) (2026)

## Abstract (arXiv)

Neural differential equations offer a powerful approach for learning dynamical systems from data. However, they do not inherently respect known constraints, such as conservation laws, that should be obeyed by the learned dynamics. It is well known that enforcing constraints in data-driven models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a method for constraining neural differential equations based on projection of the predicted velocities onto the tangent space of the manifold that fulfills the constraint. In tests on several examples from different fields, including chaotic dynamical systems and power grid models, PNDEs outperform existing methods for constraining learned dynamics, require fewer hyperparameters, and are computationally more efficient. The proposed approach demonstrates potential for enhancing the modeling of constrained dynamical systems, particularly in domains where accuracy and reliability are essential.