aiqbench analysis
This research proposes a novel architectural constraint where internal representations in learning systems strictly obey symplectic conservation laws via Legendre duality, effectively bridging information geometry with Hamiltonian dynamics. By deriving an explicit normal form for these updates and constructing Symplectic Reservoirs that preserve this structure by construction, the work offers a rigorous theoretical foundation for integrating physical phase-space principles directly into recurrent neural architectures. The study demonstrates through numerical experiments that such geometrically constrained models can be distinguished from generic baselines while maintaining stability in stochastic processes like Gaussian process regression.
Symplectic Representation of Legendre Dynamics
arXiv:2512.19409v2 Announce Type: replace 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, 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 $p = d\psi(q)$ 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.
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, 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 $p = d\psi(q)$ 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.
Machine Learning (cs.LG)
arXiv:2512.19409 [cs.LG]
(or arXiv:2512.19409v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2512.19409 Focus to learn more arXiv-issued DOI via DataCite
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