MMP.9:11 - SoTA-Echoing

The practice question is how to replace unresolved dynamics economically while retaining the needed result. Adopt construction from the source law and evaluation within the reduced evolution, including memory, initial information and question-relative outputs. A serious alternative fits an instantaneous missing term on source trajectories and judges primarily that fit. It can be cheaper and adequate for a limited use, but it can miss errors generated when the closure supplies its own future inputs.

Sanderse, Stinis, Maulik and Ahmed, Scientific machine learning for closure models in multiscale problems, version 2 (2024), sections 2.1-2.2, 3.2 and 7.1, is a comparative synthesis of closure constructions. It distinguishes an unclosed term from its replacement, and fitting the term from testing the evolving reduced model. Adapt these distinctions in :4.1-4.4. Its memory discussion supports preserving initial and historical effects during elimination. An exact elimination identity still needs an affordable way of obtaining its result; :4.2 makes that cost question explicit. The linear elimination, population bounds and transient estimates above are elementary derivations developed here, not empirical validation claims.

Freitas, Um, Desbrun, Buzzicotti and Biferale, A posteriori closure of turbulence models: are symmetries preserved? (2026 preprint), sections 3-5, provides a current countercase. A learned shell-model closure reproduces selected statistics while missing other correlations and scale-invariance properties. Adopt its consequence in :4.4: select the observables that the receiving use needs instead of extending fit of one statistic to all requested behavior. Missing memory is a proposed explanation in that case; it does not establish a universal cause or require every reduced model to carry the same memory construction.

The selected method spends effort on the omitted contribution and the use it can change; it need not reproduce every property of the detailed model. A sufficient analytic bound can be cheaper than training or testing another closure. Reopen the comparison when changed inputs, initial conditions, horizon or requested observables expose a consequential error, or when another construction obtains the same needed result at lower cost.