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LIT-1307: Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism

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StatusActive — read 2026-10-20 (NOTE-1029) in full; worth reading for the discrete-time Mori–Zwanzig identity derived from the Dyson formula in a few lines (Eq. 2.5–2.6), valid for any projection; for the Wiener projection, orthogonal projection onto the closed span of the whole past of chosen functions Ψ of the resolved variables, under which the Dyson formula collapses (Eq. 3.5), the memory term disappears into the projection, and the residual is orthogonal to all of Ψ's past (Eq. 3.6b), not only to its present (THEORY-819); for the skew-product construction that brings randomly forced systems and bounded delays under the same formalism (§3.3); and for the cascade-of-quadratics parametrization that enforces stable memory during fitting (§4.1). Read it knowing that the step from the Wiener projection to NARMAX is a rational approximation the authors themselves call "uncontrolled" and "heuristic", so no error bound and no convergence statement is proved; that the Wiener projection needs an invertible map and an invariant measure, and its claimed series representation (property (i)) needs more than the linear independence it states; that the KS and Burgers results are numerical, with orders chosen by trial for stability; that Appendix C says the noise term does not appreciably change the KS statistics or forecasts while §5.1 says that without it the models "do considerably worse at all tasks"; and that the record's check shows the rational form is exact, with orders at most n − 1, for a scalar coarse variable of a stable linear Gaussian VAR(1), where the Wiener filter's poles are the moving-average zeros, not the eliminated block's eigenvalues.
Version2
Tagsprobabilistic-modeling, mathematics, complex-systems, natural-sciences
SupportsCLAIM-397 — For a scalar stationary process with autocovariance C, the Mori–Zwanzig memory kernels for the projection onto the current value are fixed by C alone, through C(m+1) = Σ_{k≤m} c_k C(m−k); the memory vanishes iff C is geometric, and when C has finite Hankel rank d the memory has Hankel rank exactly d − 1
Cited as “paper” byNOTE-1029 — Lin and Lu — Wiener projections and the Koopman–Mori–Zwanzig formalism
Cited as “source” byTHEORY-819 — For an invertible map with an invariant measure, projecting onto the closed span of the whole past of chosen functions of the resolved variables (a Wiener projection) turns the exact discrete Mori–Zwanzig equation into one with no memory term: the next resolved value is a causal linear filter of those past functions plus a residual orthogonal to all of them, so the memory is carried by the coarse variable's own history; a finite NARMAX form follows only by a rational approximation of the filter, uncontrolled in general but exact for a scalar coarse variable of a stable linear Gaussian VAR(1)

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Lin, K. K. & Lu, F. (2021). Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism. Journal of Computational Physics 424: 109864. doi:10.1016/j.jcp.2020.109864. arXiv:1908.07725.

Read in full in NOTE-1029.

Key takeaways

  • The discrete Mori–Zwanzig equation is an exact identity for any projection, obtained by applying the Dyson formula to the observation function. It reduces nothing by itself.
  • Projecting onto the whole past of the resolved observables, rather than onto their present, removes the memory term: all memory moves into the projection, and the residual is orthogonal to every past observable. The reduced equation is then the causal Wiener filter, built only from the coarse variable’s own history (THEORY-819).
  • NARMAX is reached from there by approximating the filter’s transfer function by a ratio of polynomials. In general that step is uncontrolled. In the linear Gaussian VAR(1) class it is exact (NOTE-1029).
  • On Kuramoto–Sivashinsky (5 of 108 modes, step 0.1) and on a stochastic Burgers equation (9 of 128 modes), the fitted models reproduce spectra, autocorrelations and marginals and forecast well. That is numerical evidence for two systems, with orders chosen by trial.

Standing in the record

Filed for QUESTION-222. It supplies what Chorin and Lu (LIT-1304) left as analogy: an exact statement linking Mori–Zwanzig to a reduced model whose memory is carried by the coarse variable’s own past. It does not supply an error bound for a finite augmentation. The reading adds, for the linear class, how the Wiener filter and Mori’s kernels relate, and why their memory lengths differ.

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