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.

翻译：反向传播通常被描述为一种符号化过程：与推理在拓扑上相分离的反向传递，携带非局域误差信号，并依赖同步全局时钟——这些特征在物理实在中缺乏明确对应。现有基于物理启发的替代方案只能在微扰极限近似下恢复梯度，或需满足与前馈架构不兼容的权重对称性约束。本文通过从哈密顿最小作用量原理推导出精确反向传播，填补了这一空白。通过将前向动力学重构为连续时间形式，并针对由此产生的流调整非保守系统的拉格朗日形式体系，我们将推理与梯度计算统一至扩展相空间内的单一变分框架中，其两个共轭场分别编码激活与敏感度。单一全局拉格朗日量支配动力学：任务损失作为前向流形的对称性破缺微扰进入系统，而信用分配则展现为共轭态之间发展的张量。推理与梯度计算因此通过局域相互作用同时展开，无需独立反向电路。最终，标准反向传播被精确恢复为该连续流的离散时间投影。这一视角将物理学形式体系与反向传播相统一，为应用经典力学工具——辛几何、诺特定理、路径积分方法——分析学习动力学开辟了原则性路径。作为衍生结果，它还指向了学习内嵌于硬件本身的模拟与神经形态基板。