Physics of Learning: A Lagrangian perspective to different learning paradigms (arxiv.org)

Siyuan Guo’s arXiv paper proposes a unified, physics-inspired framework for learning: casting learning as a least-action problem and defining a "Learning Lagrangian" whose stationary trajectories correspond to efficient training dynamics. From this variational principle the author shows how familiar objects—Euler–Lagrange-style stationarity conditions—can be used to recover or reinterpret core algorithms, including Bellman’s optimality equation in reinforcement learning and the Adam optimizer in generative-model training. The central claim is that an efficient learner is one that minimizes the time (or number of observations) to reach a target error, analogous to a physical system that follows paths of least action. Technically, the work frames learning objectives as action integrals and uses variational calculus to derive update laws and optimality conditions, offering a continuous-time, principled route to algorithm design. If validated, this could unify supervised, RL and generative training under a single mathematical language, suggest new sample-efficient update rules, and connect ML practice to control theory and statistical physics (e.g., providing principled regularization or hyperparameter interpretations). The contribution is primarily theoretical: it supplies a compact explanatory lens and derivation toolkit, but its practical impact will depend on empirical validation and how the framework translates into implementable, scalable algorithms.