System-Anchored Knee Estimation for Low-Cost Context Window Selection in PDE Forecasting

Autoregressive neural PDE simulators predict the evolution of physical fields one step at a time from a finite history, but low-cost context-window selection for such simulators remains an unformalized problem. Existing approaches to context-window selection in time-series forecasting include exhaustive validation, direct low-cost search, and system-theoretic memory estimation, but they are either expensive, brittle, or not directly aligned with downstream rollout performance. We formalize explicit context-window selection for fixed-window autoregressive neural PDE simulators as an independent low-cost algorithmic problem, and propose $\mathbf{\text{System-Anchored}}\mathbf{\text{Knee}}\mathbf{\text{Estimation}}\mathbf{\text{(SAKE)}}$, a two-stage method that first identifies a small structured candidate set from physically interpretable system anchors and then performs knee-aware downstream selection within it. Across all eight PDEBench families evaluated under the shared $L\in \{1,\dots ,16\}$ protocol, SAKE is the strongest overall matched-budget low-cost selector among the evaluated methods, achieving 67.8% Exact, 91.7% Within-1, 6.1% mean regret@knee, and a cost ratio of 0.051 (94.9% normalized search-cost savings).