A Thermodynamic Theory of Learning Part II: History-Dependent Reachability and Continual Learning
Daisuke Okanohara
Sep 2026
Machine Learning
Abstract
We formulate plasticity as target-dependent, finite-horizon reachability under history-dependent dynamics, using standard minimum-energy control theory. An extended state includes parameters and internal variables that affect future updates. Around a reference trajectory, the input-to-endpoint response and its controllability Gramian determine the minimum input energy for each reachable displacement. Minimizing this energy over the joint task target defines a task-conditioned adaptation cost, exact for the arbitrary-input linear model. A solvable learning-retention example shows that this cost can increase without any rank loss, and distinguishes small directional gain from an inaccessible direction. Wasserstein transport supplies a complementary global lower bound, while learning-map Jacobians describe deformation of initial perturbations rather than response to future inputs. The resulting learning-retention frontier depends on the current extended state, admissible updates, target, horizon, and cost metric; nonlinear and constrained-update applications require additional control of these modeling choices.
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