Learning Digital Twins under Drift: Optimal Tracking Rates for Non-Stationary Dynamical Systems

We study online identification of linear-in-parameter dynamical systems whose true parameters drift over time. Measuring non-stationarity through the path variation $V_T={\sum }_{t=1}^{T-1}\Vert {\theta }_{t+1}^{\star }-{\theta }_t^{\star }\Vert$, we make two contributions. First, we give a clean blockwise dynamic-regret analysis for continuous projected online gradient descent, yielding $E[R_T]=O(T/\sqrt{\tilde{W}}+\tilde{W}{V}_T)$ and hence the oracle-tuned rate $O(T^{2/3}{V}_T^{1/3})$ . Second, we propose a residual-based adaptive step size and prove an oracle-proxy comparison theorem: when the residual proxy tracks the local prediction-error signal with lower-order overhead, the adaptive algorithm inherits the oracle scaling. Experiments on synthetic systems and real turbofan-engine degradation data (NASA C-MAPSS) support the matched-regime slope prediction on synthetic data, diagnose the failure regime when proxy overhead dominates, and show a clear unsupervised health-indicator trend in the residual proxy on FD001. A separate fast-rate guarantee under persistent excitation and a tube MPC integration are deferred to the appendix.