A Function-Space Approach to the Statistical Mechanics of Learning Dynamics

arXiv cs.AISeptember 10, 2026en

arXiv:2609.09589v1 Announce Type: new Abstract: Deep neural networks exhibit regular macroscopic behavior despite highly nonlinear dynamics in vast parameter spaces. We develop a statistical-mechanical description of learning directly in function space, treating parameter configurations as microscopic realizations and functions with their dynamical operators as macroscopic variables. For mean-squared loss, the exact error dynamics are governed by the learning operator \(M=JJ^\ast\). Combining the dynamical Boltzmann weight of the conditional stochastic dynamics with the parameter-space density of states, whose local curvature defines a statistical operator \(B\), and integrating over local f

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