# CHLU: The Causal Hamiltonian Learning Unit as a Symplectic Primitive for Deep Learning

arXiv: https://arxiv.org/abs/2603.01768 · HTML: https://mlsanity.com/paper/2603.01768/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 28 references that are in the map.
- First posted on arXiv 2 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Recent Advances in Recurrent Neural Networks](https://mlsanity.com/paper/1801.01078/) (2017)
- [Lipschitz Recurrent Neural Networks](https://mlsanity.com/paper/2006.12070/) (2020)
- [What Has a Foundation Model Found? Using Inductive Bias to Probe for World Models](https://mlsanity.com/paper/2507.06952/) (2025)
- [Denoising Hamiltonian Network for Physical Reasoning](https://mlsanity.com/paper/2503.07596/) (2025)
- [LyTimeT: Towards Robust and Interpretable State-Variable Discovery](https://mlsanity.com/paper/2510.19716/) (2025)

## Abstract (arXiv)

Current deep learning primitives dealing with temporal dynamics suffer from a fundamental dichotomy: they are either discrete and unstable (LSTMs) citeppascanudifficulty2013, leading to exploding or vanishing gradients; or they are continuous and dissipative (Neural ODEs) citepdupontaugmented2019, which destroy information over time to ensure stability. We propose the Causal Hamiltonian Learning Unit (pronounced: textitclue), a novel Physics-grounded computational learning primitive. By enforcing a Relativistic Hamiltonian structure and utilizing symplectic integration, a CHLU strictly conserves phase-space volume, as an attempt to solve the memory-stability trade-off. We show that the CHLU is designed for infinite-horizon stability, as well as controllable noise filtering. We then demonstrate a CHLU's generative ability using the MNIST dataset as a proof-of-principle.