Uniqueness in Deep Neural Networks: The Inevitable Singularity of Learning Trajectories

We establish fundamental physical and mathematical constraints in learning trajectory on parameter-level reproducibility in deep neural networks. Using dynamical systems theory, information thermodynamics, and high-dimensional geometry, we prove three impossibility theorems regarding exact structural replication of trained networks. Specifically, we demonstrate that: (i) Gradient-based training dynamics inherently exhibit positive Lyapunov exponents, leading to exponential sensitivity to initial conditions; (ii) The learning process constitutes a thermodynamically irreversible non-equilibrium process with strictly positive entropy production; (iii) In the learning process, different training trajectories occupy distinct topological equivalence classes in parameter space, as measured by persistent homology invariants. These complementary constraints collectively imply that each trained neural network traverses a structurally unique path and states with probability one, establishing inherent limits to exact reproducibility in deep learning and necessitating a paradigm shift from exact replication to distributional reproducibility and state uniqueness in machine learning science.