On the Shared Mathematics of Dynamical Systems and Neural Computation

This paper develops the thesis that the mathematics governing physical dynamical systems and the mathematics underlying contemporary artificial neural networks are not merely analogous but are, in a precise and non-metaphorical sense, expressions of the same mathematical structures. It traces this structural identity across five foundational domains: (i) differential calculus and variational optimisation, whose origins in Newton's mechanics reappear identically in gradient-based learning algorithms (ii) linear algebra and spectral theory, whose role in characterising Hamiltonian eigenstates in quantum mechanics mirrors the role of weight-matrix spectra in determining neural network expressivity and stability (iii) ordinary and partial differential equations, which simultaneously govern the time-evolution of physical fields and provide the continuous-limit interpretation of deep residual architectures (iv) variational principles and statistical mechanics, whose energy-landscape formalism furnishes the theoretical backbone of energy-based models, Boltzmann Machines, and the implicit regularisation of stochastic gradient descent; and (v) symmetry and group theory, whose role in constraining physical theories via Noether's theorem is now exploited in equivariant neural architectures for molecular simulation and beyond. The paper further examines the emerging field of Scientific Machine Learning, in which the two traditions actively hybridise: Physics-Informed Neural Networks encode differential operators directly in loss functions; Neural Ordinary Differential Equations parameterise the velocity fields of continuous dynamical systems with trainable networks; and Fourier Neural Operators learn solution maps between infinite-dimensional function spaces. It argues that this convergence is not a historical accident but reflects a structural necessity: any mathematical formalism adequate to describe organised change in complex systems will naturally serve both the physical and computational sciences.