Thermodynamic natural gradient descent (NGD-T) regulates natural-gradient steps by a geometric speed-cost bound

We introduce Thermodynamic Natural Gradient Descent (NGD-T), an optimizer that enforces a physical speed-cost constraint by combining Fisher-preconditioned updates with a dissipation-aware step-size regulator. While natural gradient methods are known to follow the steepest descent direction in information geometry, we provide a thermodynamic reinterpretation: Natural Gradient Flow uniquely minimizes instantaneous irreversible dissipation for a fixed loss decrease. NGD-T implements this principle in discrete updates by (i) preconditioning gradients with an approximate inverse Fisher, (ii) computing the geometric norm $$\:{{\Delta\:}}_{F}=\nabla\:{L}^{{\top\:}}{F}^{-1}\nabla\:L$$, and (iii) mapping a user-specified dissipation budget $$\:{Q}_{\text{b}\text{u}\text{d}\text{g}\text{e}\text{t}}$$ to a step size $$\:{\eta\:}_{T}$$ that saturates the speed-cost bound. We present numerically stable constructions for rank-deficient Fisher estimates, a hybrid nullspace fallback, and scalable K-FAC integration with eigendecomposition caching. On CIFAR-10, ImageNet, and transformer architectures, NGD-T matches or exceeds Adam in convergence while substantially reducing predicted irreversible dissipation and maintaining comparable wall-clock time. NGD-T provides a principled, tunable trade-off between learning speed and thermodynamic cost with theoretical convergence guarantees.