Compositional Dynamics in Learning and Mechanics

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arXiv:2606.28984v2 Announce Type: replace-cross Abstract: We give a single compositional setting in which gradient-based learning and Hamiltonian-style mechanics appear as functorial semantics. The syntax is an operad Arr whose objects are input-output interfaces (pairs of manifolds) and whose morphisms are *smooth adaptive arrangements*, which consist of a responsive parameter space, a lens given by smooth output and input maps, and a real-valued potential. The main technical result of the paper is what we call *lens internalization*, a lax symmetric monoidal functor Lens(C) $\to$ C associated to any symmetric monoidal closed category C. Using it, we provide two functors $\Phi_\text{phase}$, $\Phi_\text{conf}$: Arr $\to$ PC into the 2-category of polynomial coalgebras -- input-output discrete dynamical systems -- which we take as the semantics category. $\Phi_\text{phase}$ stores both position and momentum, whereas $\Phi_\text{conf}$ stores only position. When applied to a parameterized function, $\Phi_\text{conf}$ recovers the gradient descent training algorithm, with backpropagation as the lens' backward pass. When applied to harmonic particles wired together -- in series, or according to any finite directed graph -- one diagram yields two different regimes, both of which are governed by the graph Laplacian: $\Phi_\text{phase}$ gives the discrete wave equation, which is conservative and second-order, and $\Phi_\text{conf}$ gives the discrete heat equation, which is dissipative and first-order. They are two semantics of one adaptive arrangement, e.g. with the same …

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