Backpropagation as Physical Relaxation: Exact Gradients in Finite Time

Researchers have derived exact backpropagation from Hamilton's least-action principle, unifying inference and gradient computation within a single variational framework on a doubled phase space. The work, submitted to arXiv on 2 Feb 2026 and revised 11 May 2026, recasts forward dynamics in continuous time and adapts a Lagrangian formalism for non-conservative systems, enabling learning through local interactions without a separate backward circuit. This opens a principled pathway for applying classical mechanics tools to learning dynamics and points toward analog and neuromorphic hardware that embodies learning.

Computer Science Machine Learning Submitted on 2 Feb 2026 v1 https://arxiv.org/abs/2602.02281v1 , last revised 11 May 2026 this version, v2 Title:A Physical Theory of Backpropagation: Exact Gradients from the Least-Action Principle View PDF /pdf/2602.02281 HTML experimental https://arxiv.org/html/2602.02281v2 Abstract:Backpropagation is typically presented as a symbolic procedure: a backward pass topologically distinct from inference, with non-local error signals and synchronous global clocking, features with no clear analog in physical reality. Existing physics-inspired alternatives recover gradients only approximately, in vanishing-perturbation limits, or under weight-symmetry constraints incompatible with feedforward architectures. In this paper, we address this gap by deriving exact backpropagation from Hamilton's least-action principle. By recasting the forward dynamics in continuous time and adapting a Lagrangian formalism for non-conservative systems to the resulting flow, we unify inference and gradient computation within a single variational framework on a doubled phase space, whose two conjugate fields jointly encode activations and sensitivities. A single global Lagrangian governs the dynamics: the task loss enters as a symmetry-breaking perturbation of the forward manifold, and credit assignment emerges as the tension that develops between the conjugate states. Inference and gradient computation thus unfold simultaneously through local interactions, requiring no separate backward circuit. Ultimately, standard backpropagation is recovered exactly as the discrete-time projection of this continuous flow. This perspective unifies the formalism of physics with backpropagation, opening a principled pathway for applying tools from classical mechanics - symplectic geometry, Noether's theorem, path-integral methods - to the analysis of learning dynamics. As a downstream consequence, it also points toward analog and neuromorphic substrates in which learning is embodied in the hardware itself. Submission history From: Antonino Emanuele Scurria view email /show-email/80b1e1d3/2602.02281 Mon, 2 Feb 2026 16:21:05 UTC 315 KB v1 /abs/2602.02281v1 v2 Mon, 11 May 2026 09:10:38 UTC 308 KB Current browse context: cs.LG Change to browse by: References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender IArxiv Recommender What is IArxiv? https://iarxiv.org/about arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs https://info.arxiv.org/labs/index.html .