Symplectic Reservoir Representation of Legendre Dynamics
Robert Simon Fong, Gouhei Tanaka, Kazuyuki Aihara
TL;DR
The paper develops a representation-centric view of physics-inspired structure preservation by introducing Legendre dynamics, i.e., parameter updates that stay on Legendre graphs within exponential-family models. It connects this geometric invariance to symplectic geometry via a Symplectic Reservoir (SR), whose updates are symplectomorphisms arising from input-driven Hamiltonians and preserve Legendre duality at every step. A key result is the geometric characterization that Legendre-type updates are exactly cotangent lifts composed with exact fiber translations, providing a precise design space for structure-preserving representations. The framework is shown to encompass linear-time invariant Gaussian process regression and Ornstein–Uhlenbeck dynamics as Legendre dynamics, unifying these processes under a Legendre-dynamics lens and injecting symplectic geometry directly into the representation layer.
Abstract
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, conservation laws of Hamiltonian systems such as symplecticity guarantee long-term stability, and recent work has begun to hard-wire such constraints into learning models at the loss or output level. Here we ask a different question: what would it mean for the representation itself to obey a symplectic conservation law in the sense of Hamiltonian mechanics? We express this symplectic constraint through Legendre duality: the pairing between primal and dual parameters, which becomes the structure that the representation must preserve. We formalize Legendre dynamics as stochastic processes whose trajectories remain on Legendre graphs, so that 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 prove that the maps that preserve all Legendre graphs are exactly symplectomorphisms of cotangent bundles of the form "cotangent lift of a base diffeomorphism followed by an exact fibre translation". Dynamically, this characterization leads to the design of a Symplectic Reservoir (SR), a reservoir-computing architecture that is a special case of recurrent neural network and whose recurrent core is generated by Hamiltonian systems that are at most linear in the momentum. Our main theorem shows that every SR update has this normal form and therefore transports Legendre graphs to Legendre graphs, preserving Legendre duality at each time step. Overall, SR implements a geometrically constrained, Legendre-preserving representation map, injecting symplectic geometry and Hamiltonian mechanics directly at the representational level.