# Symplectic Representation of Legendre Dynamics

arXiv: https://arxiv.org/abs/2512.19409 · HTML: https://mlsanity.com/paper/2512.19409/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 42 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Builds on (most cited first, 18 of 18)

- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Reservoir Computing Universality With Stochastic Inputs](https://mlsanity.com/paper/1807.02621/) (2018)
- [Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems](https://mlsanity.com/paper/1712.00754/) (2017)
- [Universal Time-Series Representation Learning: A Survey](https://mlsanity.com/paper/2401.03717/) (2024)
- [On Invariance and Selectivity in Representation Learning](https://mlsanity.com/paper/1503.05938/) (2015)
- [Dynamical Systems as Temporal Feature Spaces](https://mlsanity.com/paper/1907.06382/) (2019)
- [Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2308.15349/) (2023)
- [A Structure-Preserving Kernel Method for Learning Hamiltonian Systems](https://mlsanity.com/paper/2403.10070/) (2024)
- [Simple Cycle Reservoirs are Universal](https://mlsanity.com/paper/2308.10793/) (2023)
- [Symmetry Preservation in Hamiltonian Systems: Simulation and Learning](https://mlsanity.com/paper/2308.16331/) (2023)
- [Universality of Real Minimal Complexity Reservoir](https://mlsanity.com/paper/2408.08071/) (2024)
- [CLPNets: Coupled Lie-Poisson Neural Networks for Multi-Part Hamiltonian Systems with Symmetries](https://mlsanity.com/paper/2408.16160/) (2024)
- [Linear Simple Cycle Reservoirs at the edge of stability perform Fourier decomposition of the input driving signals](https://mlsanity.com/paper/2412.00295/) (2024)
- [Stochastic dynamics learning with state-space systems](https://mlsanity.com/paper/2508.07876/) (2025)
- [Echoes of the Past: A Unified Perspective on Fading memory and Echo States](https://mlsanity.com/paper/2508.19145/) (2025)

## Abstract (arXiv)

Modern learning systems act on internal representations of data, yet how these representations encode underlying physical or statistical structure is often left implicit. In physics, symplecticity keeps Hamiltonian systems faithful to their phase-space geometry. Recent learning methods impose such geometric structure either in the dynamics or through training losses. Here we ask a different question: what would it mean for the representation itself to obey a symplectic conservation law? We pose this representation-level constraint through Legendre duality: the relation p = dpsi(q) between primal and dual coordinates, which in exponential family models is the information-geometric pairing of natural and expectation parameters. We formalize Legendre dynamics as stochastic processes whose trajectories remain on Legendre graphs, where the evolving primal-dual parameters stay Legendre dual. We show that this class includes linear time-invariant Gaussian process regression and Ornstein-Uhlenbeck dynamics. Geometrically, we characterize the symplectomorphisms of cotangent bundles that preserve all Legendre graphs. We show that these maps are exactly cotangent lifts of base diffeomorphisms followed by exact fibre translations. This gives an explicit normal form for Legendre-preserving representation updates. Dynamically, we prove that the normal form is realized by Hamiltonians that are at most linear in the momentum. This realization principle is used to construct linear and nonlinear Hamiltonian Symplectic Reservoirs (SR) whose recurrent updates preserve Legendre graphs by construction. This is the only normal form that preserves Legendre duality, so the architecture follows from the invariant. Numerical experiments confirm the normal-form identities and distinguish Legendre preserving Hamiltonian SRs from generic symplectic and standard reservoir baselines.