This paper develops a rigorous mathematical framework for understanding modern neural networks as infinite-dimensional dynamical systems acting on function spaces rather than on finite vectors. Neural operators—networks designed to learn mappings between spaces of functions, such as those arising in fluid mechanics or material science—are typically analysed numerically but lack a coherent theoretical foundation that connects their stability, generalization, and controllability. This study proposes a synthesis of nonlinear functional analysis, differential topology, and control theory to address this gap.
By modeling neural operators as nonlinear semigroups on Banach and Hilbert manifolds, the paper defines learning as a topological flow evolving in an infinite-dimensional state space. Training, in this formulation, is not merely optimization but a process of steering trajectories through a geometrically constrained function landscape. Stability is shown to depend not only on loss minimization but on the topological structure of attractors induced by the network architecture and regularization scheme.
The paper introduces a notion of topological generalization, where the ability of a neural operator to extrapolate across unseen physical regimes is governed by the homotopy class of its learned solution manifold. Using tools from degree theory and nonlinear spectral analysis, the study demonstrates how certain architectural choices enforce global geometric constraints that prevent chaotic overfitting even in highly underdetermined learning problems.
The control-theoretic implications are substantial. Training algorithms can be reinterpreted as feedback control laws acting on infinite-dimensional learning dynamics, opening the possibility of stability-certified and robustly controllable AI systems. This framework provides a mathematically principled route toward trustworthy machine learning for complex physical systems governed by partial differential equations.
By embedding AI within the deep structure of geometry, topology, and functional analysis, the paper reframes learning not as a black-box procedure but as a controllable mathematical process.
