Compositional Dynamics in Learning and Mechanics

[2606.28984] Compositional Dynamics in Learning and Mechanics Skip to main content Search Submit Donate Log in Mathematics > Category Theory arXiv:2606.28984 (math) [Submitted on 27 Jun 2026 ( v1 ), last revised 22 Aug 2026 (this version, v2)] Title: Compositional Dynamics in Learning and Mechanics Authors: David I. Spivak View PDF 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 potential in each case. And because Arr is an operad, such diagrams nest -- larger systems wired from smaller ones -- and each semantics assembles a system's dynamics functorially from its parts. These dynamics are moreover executable: a parameterized neural network and a graph of particles both compile, by the same construction, to explicit state machines one can run. Comments: 91 pages. Main addition is a section full of examples. Also added: a remark that almost everything works for C^1 rather than smooth, a number of suboperads (stateless, potential-free, linear, etc), a brief comparison to bond graphs Subjects: Category Theory (math.CT) ; Artificial Intelligence (cs.AI) MSC classes: 18M60, 18M35, 18M05, 18C10, 18B20, 93A13, 37N40, 37J99, 70H05, 70G99, 68T07 Cite as: arXiv:2606.28984 [math.CT] (or arXiv:2606.28984v2 [math.CT] for this version) https://doi.org/10.48550/arXiv.2606.28984 Focus to learn more arXiv-issued DOI via DataCite Submission history From: David Spivak [ view email ] [v1] Sat, 27 Jun 2026 15:39:01 UTC (91 KB) [v2] Sat, 22 Aug 2026 21:03:50 UTC (108 KB) Full-text links: Access Paper: View PDF TeX Source view license Current browse context: math.CT < prev | next > new | recent | 2026-06 Change to browse by: cs cs.AI math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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