NOTE-989: Chorin, Hald and Kupferman — Optimal prediction and the Mori–Zwanzig representation

Contribution

It joins two literatures. One is the authors’ own “optimal prediction” of underresolved computations: averaging the unresolved variables against a prior invariant measure, conditioned on the resolved ones. The other is the Mori–Zwanzig projection formalism of nonequilibrium statistical mechanics. It shows that first-order optimal prediction is the Markovian term of Zwanzig’s nonlinear generalized Langevin equation. The terms that optimal prediction drops are a memory integral and a noise orthogonal to the resolved variables. It derives the equation for any flow with an invariant measure, not only a Hamiltonian one, and shows in one example that keeping the memory term recovers a decay that the Markovian scheme misses. In the authors’ words: “The analogy between irreversible statistical mechanics and underresolved computation is the main contribution of the present paper.”

Key insight

When you follow a few functions A of a large system’s state and average over the rest, the exact equation for A has three parts. The first is the best function of A’s present value: the conditional mean of A’s rate of change given A. The second is an integral over A’s own past. The third is a term that depends on the unresolved initial state and is uncorrelated with every function of A’s initial value. A Markovian reduced model, however well chosen, keeps only the first. The error it makes over time is the irreversible loss of information about the initial data, which the memory term carries.

Assumptions

• A flow du/dt = R(u) on a phase space Γ of dimension n, “n may be infinite”, with flow map φ(u, t). The derivation is formal: no domains, no regularity, and no proof that the exponentials or S⊥ᵗ exist.
• An invariant probability measure μ⁰ on Γ, with μᵗ = μ⁰. The initial data are drawn from it, so u and A(u) are random variables. Hamiltonian structure is not needed for §2. It is used in §3’s reduced-Hamiltonian form and in §4.
• Resolved variables are a vector A = (A₁, …, A_m) of phase functions (“phase variables”). Often A = P_m u, the first m components.
• The projection is P B = E[B | A], the conditional expectation under μ⁰, read as the L²(μ⁰)-orthogonal projection onto all functions of A, not only linear ones. This is Zwanzig’s nonlinear projection, not Mori’s linear one.
• §4 only: the nonlinear Schrödinger equation iuₜ = −u_xx + ¼(3|u|²u + u*³) on [0, 2π], periodic, in Fourier modes, with the canonical measure ∝ e^{−H/T}. Everything is computed perturbatively in the quartic part H₁ about the Gaussian measure ∝ e^{−H₀/T}, by Wick’s theorem.

Key results

• Phase-variable dynamics (2.2). (SᵗA)(u) = A(φ(u, t)) satisfies ∂ₜSᵗA = LSᵗA with the Liouvillian L = Σᵢ Rᵢ(u) ∂/∂uᵢ, and SᵗL = LSᵗ, Sᵗ(AB) = (SᵗA)(SᵗB), Sᵗf(A) = f(SᵗA).

• Conditional expectation (2.3) and properties 1–5. E[B|A] is the best mean-square approximation of B by a function of A, and E[f(A)(B − E[B|A])] = 0. Cited to Chung.

• Splitting (2.4). ∂ₜSᵗA = SᵗE[LA|A] + Sᵗ(LA − E[LA|A]). With R(A) := E[LA|A] (a 𝔑 in the paper), SᵗR(A) = R(SᵗA).

• Orthogonal dynamics (2.5)–(2.6). S⊥ᵗ is defined by ∂ₜS⊥ᵗF = LS⊥ᵗF − E[LS⊥ᵗF|A], with S⊥⁰F = F. Then ∂ₜE[S⊥ᵗF|A] = 0, so if F is orthogonal to the functions of A, so is S⊥ᵗF for every t.

• Dyson formula. SᵗB = S⊥ᵗB + ∫₀ᵗ S^{t−s} E[LS⊥ˢB|A] ds. Symbolically, e^{tL} = e^{t(I−P)L} + ∫₀ᵗ e^{(t−s)L} P L e^{s(I−P)L} ds.

• Generalized Langevin equation (2.7).

  ∂ₜSᵗA = R(SᵗA) + ∫₀ᵗ K(s, S^{t−s}A) ds + S⊥ᵗF,

  with F = LA − E[LA|A] and K(s, A) = E[LS⊥ˢF | A]. “An identity between phase variables.” The first term is Markovian. The second is memory, depending on SˢA for 0 ≤ s ≤ t. The third is “noise”: it needs the full initial u, and E[S⊥ᵗF | A] = 0 for all t. The authors note that this is “a weaker property than statistical independence”.

• First-order optimal prediction (3.1). It approximates E[SᵗA | A] by v(t), where dv/dt = R(v) and v(0) = A(u). That is (2.7) with the memory and noise dropped.

• (3.2), resolved components. For A = P_m u, R(P_m u) = E[P_m R(u) | P_m u], the ratio of integrals over u_{m+1}, …, u_n against the density. This is contrasted with Galerkin truncation, dv/dt = P_m R(v₁, …, v_m, 0, …, 0).

• Hamiltonian case. With the canonical density Z⁻¹e^{−H}, the resolved pairs (Q, P) obey dQᵢ/dt = ∂𝓗/∂Pᵢ and dPᵢ/dt = −∂𝓗/∂Qᵢ, where 𝓗(Q, P) = −log ∫ e^{−H} dp_{m+1}⋯dp_n dq_{m+1}⋯dq_n is the “reduced, or renormalized, Hamiltonian”. This was re-derived here by differentiating under the integral.

• §4, Markovian term (4.4). It is truncation plus a frequency shift, ℜ_k ≈ −i(ω_k + C)u_k − (i/4) Σ (u_{k₁}u{k₂} + 3u{−k₁}u_{−k₂}) u*{k₃} δ{k,−k₁−k₂−k₃}, with C = (3T/2π) Σ_{|k|>m} 1/ω_k and ω_k = k². Checked: C ≈ 3T/(πm), “like 1/m”, and C = π/2 at m = 0 and T = 1, which matches (4.11).

• §4, noise and memory (4.5)–(4.9). To leading order the noise is (I − P)S₀ᵗL₁u_k, using (I − P)L₀P = 0, so that S⊥,₀ᵗ(I − P) = (I − P)S₀ᵗ. The memory kernel is quadratic in L₁. It is a sum of terms linear, cubic and quintic in the resolved modes, with explicit oscillatory coefficients κ_q, φ_q and ψ_q. At m = 0, (4.11) keeps only the κ₀ and φ₀ terms. Checked: ψ₀ ≡ 0, and the 3/16 terms need |k₃| > 0 with k₃ = −j₁ − j₂ − j₃ = 0, so they vanish.

• Mean equation (4.10). dv/dt = ℜ(v(t)) + ∫₀ᵗ K(s, v(t − s)) ds, obtained by averaging (2.7) conditionally, which removes the noise exactly, and then replacing E[ℜ(SᵗA)|A] by ℜ(v) and E[K(s, ·)|A] by K(s, v). The second step is a mean-field closure and is not controlled.

• Figure 1. m = 0, T = 1, with a Monte Carlo mean over 10⁴ solutions conditioned on u₀. First-order optimal prediction oscillates without decay at amplitude ≈ 0.03. The memory equation with κ₀ and φ₀ “truncated at t = 0.9”, a cut-off “chosen so as to produce a reasonable decay”, decays. It follows the Monte Carlo curve to t ≈ 2. After that it decays more slowly and drifts in phase: at t ≈ 6 it peaks near 0.012, while the Monte Carlo mean is near 0.

Claims

id	claim	strength	support
C1	For any flow with an invariant measure, the resolved variables obey the generalized Langevin identity (2.7), with conditional expectation as the projection	strong as a formal identity	Dyson formula “checked by differentiation”; the derivation in §2. No functional-analytic conditions are given. Invariance is assumed but is not used in deriving (2.7) itself (see Limitations)
C2	The noise S⊥ᵗF has zero conditional mean given the initial A for all t, so it drops out of the conditional-mean equation	strong	(2.6)
C3	First-order optimal prediction is (2.7) without memory and noise	strong	comparison of (2.7) with (3.1)
C4	”No Markovian approximation of SᵗA by a system of m equations can be more accurate, in the mean square sense, than first order optimal prediction”	overstated	(2.3) gives this for the instantaneous rate of change, not for trajectories over time; a counterexample is in Limitations
C5	For a canonical Hamiltonian system, first-order optimal prediction is Hamiltonian in the renormalized 𝓗 = −log∫e^{−H}	strong	derivation in §3, re-derived here
C6	First-order optimal prediction works for short times, for m not too small and at low temperature, and fails for severely underresolved, hot, long runs	weak here	asserted; supported by earlier papers [2–4], not by this one
C7	Including memory “is crucial” for capturing the relaxation of the mean, and gives means “faithful to the true means for longer times”	moderate for “captures the decay”, weak for “faithful”	one example: m = 0, perturbative kernels, cut off at a time chosen to give a reasonable decay; Figure 1 shows phase and amplitude errors after t ≈ 2
C8	Loss of accuracy in underresolved computation is analogous to the loss of information in irreversible statistical mechanics, as the conditioned measure relaxes to the invariant one	interpretive	§5; the formal content is C1–C3

Method

• Write the resolved variables’ evolution in the Heisenberg form SᵗA.
• Split its rate of change by the conditional-expectation projection (2.4).
• Rewrite the orthogonal part with the Dyson formula. That gives (2.7).
• For a concrete system, expand L = L₀ + L₁ about an integrable quadratic part. Compute R, the noise and K to leading order with Wick’s theorem under the Gaussian measure of H₀. Average (2.7) conditionally, close it by evaluating R and K at the conditional mean, and integrate the resulting integro-differential equation (4.10).

Concepts

• Phase variable: a function A on phase space. SᵗA is its value along the flow.
• Liouvillian: L = Σᵢ Rᵢ ∂/∂uᵢ, the generator of Sᵗ = “e
• Orthogonal dynamics: S⊥ᵗ, the evolution generated by (I − P)L. It keeps functions orthogonal to the span of functions of A.
• Memory function: K(t, A) = E[LS⊥ᵗF | A], a function of A that varies in t.
• Optimal prediction: estimating the mean future of the resolved variables, E[SᵗA | A], under a prior invariant measure. “First-order” is its Markovian approximation (3.1). “Optimal” refers to (2.3): R is the best mean-square function of A for the instantaneous rate.
• Renormalized Hamiltonian: 𝓗(Q, P) = −log∫e^{−H} over the unresolved pairs, “minus the logarithm of the weighted phase space volume constrained by the known values”.
• Temperature: T in the canonical density, which “specifies the variance of the samples”.

Connections

It rests on Mori (1965) and Zwanzig’s nonlinear generalization (1980), following Evans and Morriss’s presentation of the Dyson formula. It continues Chorin, Kast and Kupferman’s first-order optimal prediction (1998, 1999), Chorin, Kupferman and Levy’s nonlinear Schrödinger study (LBNL-44105), and Hald’s rigorous analysis (1999), which is cited but not reproduced. The “later work” it promises, on memory truncation with more resolved variables and on sampling-based higher-order schemes, is the programme of Chorin, Hald and Kupferman’s “Optimal prediction with memory” (LIT-1253, registered unread). Parish and Duraisamy (LIT-1259, unread) apply the formalism to large-eddy closure.

Bearing on the record

This is the record’s first reading of Mori–Zwanzig. What follows the first item is the record’s own derivation, not the paper’s. The scripts are in the session scratchpad. None of it is filed as a THEORY.

• QUESTION-222, the general form. A3’s Ż = F(Z) + ∫K(t − s, Z_s) ds + η is (2.7). The paper fixes three things A3 left open. The projection is conditional expectation under an invariant measure, so it is not arbitrary once the measure and A are chosen. The noise is a deterministic function of the unresolved initial state, uncorrelated with functions of A(u) but not independent of them. And the kernel K(s, A) is in general nonlinear in A. A3’s caution holds within the paper: change A or μ⁰, and R, K and F all change.

• Fluctuation–dissipation (the record’s derivation). The paper relates K to the noise only by the definition K = E[LS⊥ˢF|A]. A sharper relation follows from its own assumption of invariance. If μ⁰ is invariant, L is skew-adjoint on L²(μ⁰), because E[fLg] + E[gLf] = E[L(fg)] = (d/dt)E[Sᵗ(fg)] = 0 at t = 0. Take G = S⊥ˢF, which is orthogonal to the functions of A for every s, and any test function h. Then E[h(A)K(s, A)] = E[hLG] = −E[G·Lh(A)] = −E[G ∇h(A)·F], since ∇h·R(A) is a function of A. With Mori’s linear projection this gives K(s) = −E[S⊥ˢF Fᵀ] C_A⁻¹. That is the second fluctuation–dissipation theorem, checked numerically to 10⁻¹⁵ for a random skew-symmetric linear flow with m = 2 and n = 6, together with the Volterra form of (2.7). With h = Aᵢ it gives E[Aᵢ K_j(0, A)] = −E[Fᵢ F_j] for the nonlinear projection too.

• So, in the paper’s setting, memory and noise vanish together (the record’s). By the last identity, K(0, ·) ≡ 0 forces E[|F|²] = 0, and F = 0 makes K ≡ 0 and the noise ≡ 0. For a deterministic flow with an invariant measure, then: zero memory ⇔ zero noise ⇔ F = 0 ⇔ dA/dt = R(A) exactly, which is closure. Whenever first-order optimal prediction is not exact, the memory kernel is nonzero. For a linear flow ẋ = Jx and A = Lx, F = 0 is LJ = BL. That is Shpak et al.’s ĀQ = QA (NOTE-979) and the continuous-time form of THEORY-733’s L A = B L. This was checked for an invariant 2-plane of a skew J (|F| ~ 10⁻¹⁵).

• THEORY-733 and THEORY-257: MZ memory against dynamical independence in a linear Gaussian VAR(1) (the record’s). Barnett and Seth’s setting is stochastic and in discrete time, so the paper’s FDT does not carry over. Apply the discrete Dyson identity 𝒦^{t+1} = Σ_{j=0}^{t} 𝒦^{t−j} P𝒦(Q𝒦)^j + (Q𝒦)^{t+1} to the transition operator 𝒦g(x) = E[g(X_{t+1}) | X_t = x]. Take P = E[· | Y] under the stationary law N(0, Π), with Y = LX and Q = I − P. On linear observables, 𝒦 acts as c ↦ Aᵀc, and P is the Π-orthogonal projection onto row(L), which for Gaussians is the full conditional expectation. In a Π-orthonormal basis with row(L) first, write 𝒦 = [[K₁₁, K₁₂], [K₂₁, K₂₂]]. Then the Markov term is K₁₁ = C_Y(1)C_Y(0)⁻¹, the memory kernels are M_j = K₁₂K₂₂^{j−1}K₂₁ for j ≥ 1, and the first orthogonal-dynamics term is K₂₁. The Volterra identity was checked to 10⁻¹⁷. Then:

  1. Exact dynamical independence ⇔ K₂₁ = 0. Q𝒦P = 0 iff Aᵀ maps row(L) into itself, iff LA = BL, which is THEORY-733’s criterion. In words, E[Y_{t+1} | X_t] is a function of Y_t. In the MZ split, DI is the vanishing of the noise from the unresolved initial state, not of the memory.
  2. DI ⇒ zero memory. Every M_j contains K₂₁. Checked: a left eigenvector of A gives |K₂₁|, max|M_j| and T(X → Y) all ~ 10⁻¹⁶.
  3. Zero memory does not imply DI. Zero memory ⇔ C_Y(t) = K₁₁ᵗ C_Y(0) for all t ⇔ Y is Markov in its own filtration. Counterexample: A = [[0.5, 0.4], [0, 0.3]], Σ = I − AAᵀ (so Π = I), L = e₁. Then C_Y(t) = 0.5ᵗ exactly and every M_j = 0, yet T(X → Y) = ½log((1 − a₁₁²)/(1 − a₁₁² − a₁₂²)) = 0.120 nats, in closed form and numerically. In the Π-orthonormal frame, K₂₁ is the inward leg (the rest driving Y), K₁₂ the outward leg, and the memory is the round trip Y → rest → Y. DI asks only that the inward leg vanish, which matches THEORY-257’s reading (“dynamical independence asks only for none coming in”). Zero memory needs the round trip to vanish, which an absent outward leg also achieves. The legs are measured in Π-whitened coordinates, not in raw entries of A. Zero memory is non-generic: none of 200 random 3-dimensional models with m = 1 had it.
  4. For VAR(p ≥ 2), DI does not imply zero memory either, with the projection onto the current Y_t. In a VAR(2) where x₁ is an undriven AR(2), T(X → Y) = 0 and max|M_j| = 0.35. So A5’s caution, that non-Markovianity is not a failure of dynamical independence, is right for p ≥ 2. For p = 1 it is half right: there nonzero memory does imply dependence.
  5. QUESTION-222’s own example shows item 3 in continuous time. In ẋ = −ax + by, ẏ = cx − dy, K = bc e^{−dt}. With c = 0 and b ≠ 0 the memory vanishes, but x is not closed: b e^{−dt} y(0) remains. That system is dissipative and has no invariant measure with a skew-adjoint L, which is why the paper’s setting excludes it.
  6. Reversible VAR(1) restores the paper’s picture. If the chain satisfies detailed balance (AΠ symmetric), 𝒦 is Π-self-adjoint, so K₁₂ = K₂₁ᵀ and M₁ = K₂₁ᵀK₂₁, with tr M₁ = ‖K₂₁‖²_F. That is the discrete counterpart of the FDT. There, M₁ = 0 forces K₂₁ = 0: zero memory ⇔ exact dynamical independence. Checked on three random reversible models with n = 5 and m = 2 (|K₁₂ − K₂₁ᵀ| ~ 10⁻¹⁵). Counterexample 3 is not reversible. The gap between memory and dependence is a feature of irreversible (driven, one-way) coupling.

  The precise relation for a stable VAR(1) with Gaussian innovations and a linear coarse-graining, by derivation and numpy check: exact dynamical independence ⇔ the first MZ orthogonal-dynamics term Q𝒦P vanishes ⇒ the memory kernel vanishes; the converse fails on a non-generic set; and for p ≥ 2, or with the projection onto the instantaneous Y, the two properties are logically independent. For a reversible VAR(1), zero memory and DI coincide. In the paper’s own setting, a deterministic flow with an invariant measure, memory, noise and non-closure vanish together.

• QUESTION-222, augmentation (the record’s, linear class). There are two distinct “Markovian augmentations”, and they differ in size.

  • Making Y’s own law Markov needs k = rank of the Hankel matrix of (M_j) memory coordinates. That is the minimal realization of (K₂₂, K₂₁, K₁₂).
  • Making the augmented variable exactly dynamically independent needs the Krylov space of Aᵀ on row(L), that is, m plus the reachable dimension of (K₂₂, K₂₁).
  • Both are exact, and both are generically the whole eliminated dimension. A random case with n = 5 and m = 1 gave Hankel rank 4 and Krylov dimension 5. A driven chain gave 2 and 3, and the Krylov augmentation was checked to have T = 0. In counterexample 3 they are 0 and 2.
  • So “add memory coordinates” answers the question only if it says which closure is meant. Controlled-error truncation, the question’s real demand, is not touched by this paper.

• QUESTION-221 (a by-product, the record’s). In a VAR(1) whose block Y receives coupling A_{YR} from the rest: T(X → Y) ≤ ½ log det(I + (LΣLᵀ)⁻¹ A_{YR} Cov(R_t | Y_t) A_{YR}ᵀ). The reason is that conditioning on Y’s whole past does no worse than conditioning on Y_t. So the dynamical dependence is O(ε²) in the inward coupling, whatever the outward coupling. Checked: T/ε² ≈ 0.53 for ε from 0.2 to 0.025, under the bound throughout, with correlated noise. This answers the forward half of the question for linear Gaussian VAR(1) with Cov(R | Y) bounded. The MZ memory in the same runs is O(ε), not O(ε²), so memory is the coarser signal.

• CLAIM-327 and CLAIM-331. Reading 3 says how a variable members read from the past can be closed without being Markov in its own history, or Markov without being closed. The paper says nothing about timescales beyond “short times”, “m not too small” and “low temperature”. It gives no support to the idea that slower variables have less memory.

• CLAIM-388 and THEORY-755. They are consistent with readings 1 and 2. An exactly closed collective mode, K₂₁ = 0, carries no MZ memory and no noise from the eliminated state. Only its own innovation Lε remains.

No instruction for machine-learning practice; nothing for the anthology.

Limitations

• C4 is overstated. (2.3) makes R(A) = E[LA|A] the best mean-square function of A for the instantaneous rate LA. It does not make the integrated trajectory v(t) the best m-dimensional Markovian predictor of SᵗA. Counterexample, checked analytically and by Monte Carlo with 2 × 10⁵ samples: a harmonic oscillator q’ = p, p’ = −q, with canonical measure N(0, I) and A = q. Then R(q) = E[p|q] = 0, so first-order optimal prediction gives v(t) = q₀. Its mean-square error is 2 − 2cos t, which is 4 at t = π. The one-equation Markovian model v’ = −cv with c large has error close to 1 at t = π. At t = 0.5 the best c ≈ 0.26 beats it too: 0.230 against 0.245. The claim holds at t → 0⁺ only.
• Invariance is assumed more widely than it is used. The Dyson formula and (2.7) are algebraic identities for any projection P, including conditional expectation under a non-invariant μ⁰. Invariance matters for reading E[SᵗA | A] as a prediction under the prior, for the noise’s statistics, and for the fluctuation–dissipation relation above, which the paper does not state.
• No validity criterion for the Markovian approximation. “Conditions under which this is legitimate are discussed in [1–5].” Here the only conditions are short times, enough resolved modes and low temperature. The FDT above implies that K(0) is the size of the noise covariance, so memory is small exactly when F is. The paper does not say this.
• The demonstration in §4 is thin. It uses m = 0 only, a perturbation series the authors say is “not likely to be accurate for long times”, a Markovian term “not likely to be accurate when m = 0”, a mean-field closure in (4.10), and kernels cut off at t = 0.9, a time “chosen so as to produce a reasonable decay”. The cut-off is fitted to the outcome it is then said to reproduce. Figure 1 shows the decay but not a faithful mean after t ≈ 2.
• The base Gaussian measure is improper in the k = 0 mode. ω₀ = 0, so the stated conditional variance (πω_k)⁻¹T is infinite there. Conditioning on u₀ when m = 0 makes this harmless for (4.11), but it is not remarked.
• Typos in this copy. “P = (P₁, …, Q_m)” in §3. “No general way of solving equation (2.6)”, which should be (2.5). “From equation (4.8) it follows”, which reads as (2.6). The print version may have corrected them.
• Formal throughout. No existence for S⊥ᵗ, no domain for L, and no treatment of n = ∞, which is deferred to Hald (1999).
• Copy. The authors’ preprint, not the PNAS typeset text.

Open questions

• When is the memory kernel short enough that a truncation, or a few augmented coordinates, gives bounded error over a stated horizon? The paper promises “later work” (see LIT-1253). For the linear class, balanced truncation of (K₂₂, K₂₁, K₁₂) would give an error bound in terms of Hankel singular values. That would answer QUESTION-222’s demand for k and the error in terms of the eliminated spectrum.
• Does the record’s VAR(1) relation (DI ⇔ Q𝒦P = 0 ⇒ zero memory) extend to near-independence? Is the dynamical dependence bounded by a norm of the orthogonal-dynamics term, as the O(ε²) bound suggests, while the memory norm only bounds it from one side?
• For an irreversible VAR(1), can the gap between zero memory and dynamical independence be bounded by the antisymmetric part of 𝒦 in the Π-whitened frame, which measures one-way coupling? Item 6 settles only the reversible end.