# Semi-Explicit Neural DAEs: Learning Long-Horizon Dynamical Systems with Algebraic Constraints

arXiv: https://arxiv.org/abs/2505.20515 · HTML: https://mlsanity.com/paper/2505.20515/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 31 references that are in the map.
- First posted on arXiv 26 May 2025.

## Builds on (most cited first, 14 of 14)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [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)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Challenges in Training PINNs: A Loss Landscape Perspective](https://mlsanity.com/paper/2402.01868/) (2024)
- [Fashionable Modelling with Flux](https://mlsanity.com/paper/1811.01457/) (2018)
- [Symplectic Learning for Hamiltonian Neural Networks](https://mlsanity.com/paper/2106.11753/) (2021)
- [Hamiltonian neural networks for solving equations of motion](https://mlsanity.com/paper/2001.11107/) (2020)
- [Deconstructing the Inductive Biases of Hamiltonian Neural Networks](https://mlsanity.com/paper/2202.04836/) (2022)
- [Differentiable Programming for Differential Equations: A Review](https://mlsanity.com/paper/2406.09699/) (2024)
- [Constants of motion network](https://mlsanity.com/paper/2208.10387/) (2022)
- [Unifying physical systems' inductive biases in neural ODE using dynamics constraints](https://mlsanity.com/paper/2208.02632/) (2022)

## Built on by (most cited first, 4 of 4)

- [Hard-Constrained Neural Networks with Physics-Embedded Architecture for Residual Dynamics Learning and Invariant Enforcement in Cyber-Physical Systems](https://mlsanity.com/paper/2511.23307/) (2025)
- [Physics-Guided Dimension Reduction for Simulation-Free Operator Learning of Stiff Differential-Algebraic Systems](https://mlsanity.com/paper/2604.19930/) (2026)
- [Jacobian Regularization Stabilizes Long-Term Integration of Neural Differential Equations](https://mlsanity.com/paper/2602.04608/) (2026)
- [An Invariant Compiler for Neural ODEs in AI-Accelerated Scientific Simulation](https://mlsanity.com/paper/2603.23861/) (2026)

## Abstract (arXiv)

Despite the promise of scientific machine learning (SciML) in combining data-driven techniques with mechanistic modeling, existing approaches for incorporating hard constraints in neural differential equations (NDEs) face significant limitations. Scalability issues and poor numerical properties prevent these neural models from being used for modeling physical systems with complicated conservation laws. We propose Manifold-Projected Neural ODEs (PNODEs), a method that explicitly enforces algebraic constraints by projecting each ODE step onto the constraint manifold. This framework arises naturally from semi-explicit differential-algebraic equations (DAEs), and includes both a robust iterative variant and a fast approximation requiring a single Jacobian factorization. We further demonstrate that prior works on relaxation methods are special cases of our approach. PNODEs consistently outperform baselines across six benchmark problems achieving a mean constraint violation error below 10-10. Additionally, PNODEs consistently achieve lower runtime compared to other methods for a given level of error tolerance. These results show that constraint projection offers a simple strategy for learning physically consistent long-horizon dynamics.