Analytic and Variational Stability of Deep Learning Systems

We propose a unified analytic and variational framework for studying stability in deep learning systems viewed as coupled representation-parameter dynamics. The central object is the Learning Stability Profile, which tracks the infinitesimal response of representations, parameters, and update mechanisms to perturbations along the learning trajectory. We prove a Fundamental Analytic Stability Theorem showing that uniform boundedness of these stability signatures is equivalent, up to norm equivalence, to the existence of a Lyapunov-type energy that dissipates along the learning flow. In smooth regimes, the framework yields explicit stability exponents linking spectral norms, activation regularity, step sizes, and learning rates to contractivity of the learning dynamics. Classical spectral stability results for feedforward networks, a discrete CFL-type condition for residual architectures, and parametric and temporal stability laws for stochastic gradient methods arise as direct consequences. The theory extends to non-smooth learning systems, including ReLU networks, proximal and projected updates, and stochastic subgradient flows, by replacing classical derivatives with Clarke generalized derivatives and smooth energies with variational Lyapunov functionals. The resulting framework provides a unified dynamical description of stability across architectures and optimization methods, clarifying how architectural and algorithmic choices jointly govern robustness and sensitivity to perturbations. It also provides a foundation for further extensions to continuous-time limits and geometric formulations of learning dynamics.