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LIT-1304: Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics

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Properties6
StatusActive — read 2026-10-20 (NOTE-1028) in full; worth reading for the fully discrete stochastic parametrization: fix a numerical integrator for the resolved equations, take the discrepancy z^{n+1} = (x^{n+1} − x^n)/δ − R_δ(x^n), which the data give without differentiation, and fit it as a NARMAX time series with x as exogenous input, so that memory of past x, past z and past innovations is carried by finitely many lags; for the conditional-likelihood (least-squares) fit; and for a Lorenz 96 test with no time-scale separation (ε = 0.5), where one or two lags reproduce the pdf, autocorrelation and cross-correlation of x₁ and, at δ = 0.05, double the forecast lead time over a polynomial-plus-AR(1) closure (THEORY-818). Read it knowing that the link to Mori–Zwanzig is motivation, not derivation: nothing shows that the fitted Φ approximates the MZ memory or that ξ is the MZ noise, which is orthogonal to the resolved variables but neither white nor independent; that "generalizes the MZ formalism" is an overstatement; that the model structure is found "by trial and error" and changes with δ; that nothing is proved, no error bound is given and no case is shown in which a finite discrete memory suffices exactly; and that the δ = 0.01 fit has a moving-average coefficient 0.9997, at the edge of invertibility.
Version2
Tagsprobabilistic-modeling, complex-systems, mathematics, natural-sciences
SupportsCLAIM-398 — In a stable linear Gaussian VAR(1) on ℝⁿ, a scalar linear coarse variable Y = LX is, in its own law, an ARMA process of orders at most (n, n − 1); so its Mori–Zwanzig memory, though non-zero at every lag in general, is carried exactly by finitely many coordinates that are functions of Y's own past and innovations
Cited as “paper” byNOTE-1028 — Chorin and Lu — Discrete stochastic parametrization by NARMAX
Cited as “source” byTHEORY-818 — In two-scale Lorenz 96 without time-scale separation (ε = 0.5, K = 18, J = 20, F = 10), a discrete-time closure for each resolved variable that fits the one-step discrepancy of a fourth-order Runge–Kutta step as a NARMAX series, with one autoregressive lag, at most two lags of the resolved variable and at most one moving-average term, reproduces the mean, spread, distribution, autocorrelation and neighbour cross-correlation of x₁ (Kolmogorov–Smirnov distance 0.005) and, with observations every 0.05, extends the 20-member ensemble forecast lead time at anomaly correlation 0.6 from about 2 to about 4 time units over a polynomial-plus-AR(1) closure; the model structure was chosen by trial and error, differs between the two observation spacings, and carries no error bound

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Chorin, A. J. & Lu, F. (2015). Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics. Proceedings of the National Academy of Sciences 112(32): 9804–9809. doi:10.1073/pnas.1512080112. arXiv:1503.08766 (“A discrete approach to stochastic parametrization and dimensional reduction in nonlinear dynamics”).

Read in full in NOTE-1028.

Key takeaways

  • Working in discrete time removes the need to differentiate data: the discrepancy between the data and one step of a chosen integrator is exact, and it absorbs the integrator’s error together with the unresolved variables’ effect.
  • A NARMAX model of that discrepancy is a finite-dimensional Markovian augmentation of the resolved variable: its state is a few past values of x, of z and of the innovations ξ. The moving-average terms let finitely many coordinates carry a memory that is non-zero at every lag.
  • On Lorenz 96 without scale separation, one or two lags suffice to reproduce the resolved statistics and to improve forecasts (THEORY-818). That is shown numerically for one system, not proved.
  • The relation to Mori–Zwanzig is an analogy. The paper neither derives NARMAX from the MZ identity nor bounds the error of the fitted model.

Standing in the record

Filed for QUESTION-222. It is the record’s first reading of a discrete, data-driven closure that carries memory in a finite set of lags, beside the analytical Mori–Zwanzig readings (LIT-1245, LIT-1253) and the turbulence closure of LIT-1259 (THEORY-760). It supplies no bound. The record’s own check in NOTE-1028 shows, for the linear Gaussian class, why a finite discrete representation can be exact where the memory kernel never vanishes.

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