Dynamics, Stability, and Generative Flows for AI: State-Space Dynamics, Lyapunov Stability, Stochastic Evolution, and Generative Transport

Modern AI systems are not static functions. Recurrent and state-space models carry information across long contexts; neural ODEs and equilibrium layers compute through dynamical evolution; world models are rolled out over time; diffusion and flow-matching models transport probability laws from noise to data. What matters in deployment is therefore not only what a model learns at one step, but what happens when it is iterated, integrated, sampled, or placed in a feedback loop.

Dynamics, Stability, and Generative Flows for AI develops a unified mathematical language for these systems. It moves from states, maps, flows, invariant structures, Lyapunov stability, contraction, and finite-time propagation to bifurcations, chaos, stochastic evolution, mixing, and metastability. Along the way, it shows how perturbations, memories, and gradients propagate through learned dynamics, and why local stability, long-horizon behaviour, and numerical implementation must be analysed separately.

The final part applies this framework to recurrent and state-space layers, continuous-depth and equilibrium networks, normalizing flows, diffusion models, flow matching, stochastic interpolants, and long-horizon learned systems. Throughout, the book follows the full model lifecycle—architecture, training, deployment, and verification—and distinguishes the learned mathematical object from the law actually delivered by a numerical solver or sampler.

Written for graduate students, researchers, and mathematically oriented practitioners, the book combines rigorous theorem statements with geometric intuition, worked examples, computational diagnostics, and exercises. Its central message is practical as well as theoretical: reliable AI requires stability, finite-time amplification, stochastic evolution, numerical error, and distributional error to be studied as parts of one dynamical system.