On data-driven parameterizations of multidimensional generalized Langevin dynamics in the presence of a quadratic potential

What the math gives to ML

The paper gives a constructive route from empirical velocity autocorrelations to a finite-dimensional Markovian system with auxiliary variables, avoiding explicit inversion of a memory kernel. Its transferable asset is a stable Prony/Ornstein–Uhlenbeck realization: long-range temporal dependence is represented by a small bank of exponentially decaying modes, while the block structure preserves a physically meaningful covariance and can enforce stability. In neural networks, this suggests replacing quadratic-cost temporal attention or unconstrained recurrent memory with data-fitted multi-timescale state-space filters. The most direct experiment is to fit decay modes from feature autocorrelations and compare an O(TK) memory layer against attention or an unconstrained SSM at equal parameter count.