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arxiv: 2606.23341 · v1 · pith:GJ35IDS3new · submitted 2026-06-22 · 🧮 math.DS

Partial Observation of Linear Systems with the Mori-Zwanzig Formalism

Fan Wang , Peter Benner , Jan Heiland This is my paper

Pith reviewed 2026-06-26 06:41 UTC · model grok-4.3

classification 🧮 math.DS
keywords Mori-Zwanzig formalismreduced order modelinglinear time-invariant systemspartial observationsKoopman generatormemory effectsvariation of constants formula
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The pith

For linear time-invariant systems the Mori-Zwanzig formalism gives closed-form Markovian, memory, and noise terms that recover variation-of-constants reduced dynamics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an explicit formulation of the Mori-Zwanzig equation for linear time-invariant systems under partially observed observables. By expressing the dynamics in terms of observables, the Koopman generator, and projections onto resolved and unresolved components, it derives closed-form representations of the Markovian, noise, and memory contributions. These formulas recover the reduced dynamics from the variation-of-constants formula while retaining the operator-based structure. This serves as a transparent reference case for reduced-order modelling with memory and clarifies how unresolved variables influence the observed dynamics through history-dependent terms.

Core claim

By expressing the dynamics in terms of observables, the Koopman generator, and projections onto resolved and unresolved components, we derive closed-form representations of the Markovian, noise, and memory contributions that arise in the Mori-Zwanzig identity. For the linear setting, the resulting formulas recover the reduced dynamics obtained from the variation-of-constants formula while retaining the operator-based structure of the Mori-Zwanzig approach.

What carries the argument

The Mori-Zwanzig identity applied using the Koopman generator and projections onto resolved and unresolved components to obtain explicit expressions for the memory, noise, and Markovian terms.

If this is right

  • The formalism produces interpretable reduced-order models for partially observed linear systems.
  • Unresolved variables affect the observed dynamics through explicit history-dependent memory terms.
  • The derivation identifies the components needed to extend the approach to nonlinear systems and general projections including spectral filtering.
  • Examples with the harmonic oscillator and wave equations demonstrate the construction and its use for interpretable models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Data-driven approximations of the memory effects could be developed using this explicit structure as a guide.
  • The projection method may extend to problems in control or estimation with partial state information.
  • Numerical verification on additional linear systems would confirm if the closed forms hold beyond the presented examples.
  • The operator structure might allow combination with other model reduction techniques for linear systems.

Load-bearing premise

The system is linear time-invariant, allowing the Koopman generator and projections to produce closed-form expressions for the memory, noise, and Markovian terms.

What would settle it

Computing the memory term from the Mori-Zwanzig derivation and comparing it to the unresolved part from the variation-of-constants formula on a linear system such as the harmonic oscillator; mismatch would falsify the recovery claim.

Figures

Figures reproduced from arXiv: 2606.23341 by Fan Wang, Jan Heiland, Peter Benner.

Figure 1
Figure 1. Figure 1: Trajectories of the reference solution (solid) and MZ solution (dashed) for the six observed state variables of the damped oscillator chain under partial observation (ms = 3 of Ns = 5 oscillators observed). Damped Oscillatory System. We consider a chain of Ns = 5 oscillators with uniform mass m = 1.0 kg, spring constant k = 1.0 N/m, and damping coefficient c = 0.1 N s/m. The system is discretized in time u… view at source ↗
Figure 2
Figure 2. Figure 2: The results of the 2D wave equation modelled by MZ with some parts (in gray) be￾ing unobserved. The left panel illustrates the reference solution, while the right panel demonstrates the future prediction of the observed variables using the reconstructed MZ model. The upper row shows the initial state in the simulation, and the lower row shows the state at final time. Preprint (Max Planck Institute for Dyna… view at source ↗
read the original abstract

The Mori-Zwanzig formalism provides a systematic framework for deriving reduced-order model of dynamical systems when only part of the state is observed, but its practical use is often limited by the complexity of the resulting computations. This paper develops an explicit formulation of the Mori-Zwanzig equation for linear time-invariant systems under partially observed observables. By expressing the dynamics in terms of observables, the Koopman generator, and projections onto resolved and unresolved components, we derive closed-form representations of the Markovian, noise, and memory contributions that arise in the Mori-Zwanzig identity. For the linear setting, the resulting formulas recover the reduced dynamics obtained from the variation-of-constants formula while retaining the operator-based structure of the Mori-Zwanzig approach. This makes the derivation a transparent reference case for reduced-order modelling with memory and clarifies how unresolved variables influence the observed dynamics through history-dependent terms. The analysis also identifies the ingredients needed for extensions to nonlinear systems and more general projections, including spectral filtering and data-driven approximations of memory effects. Analytical and numerical examples involving the harmonic oscillator and wave equations illustrate the construction and demonstrate how the formalism can be used to obtain interpretable reduced-order models for partially observed systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper develops an explicit formulation of the Mori-Zwanzig (MZ) equation for linear time-invariant systems under partial observations. Using observables, the Koopman generator, and orthogonal projections onto resolved and unresolved components, it derives closed-form expressions for the Markovian, noise, and memory terms. These recover the reduced dynamics from the variation-of-constants formula while preserving the operator structure of MZ, with examples on the harmonic oscillator and wave equations to illustrate interpretable reduced-order models.

Significance. If the derivations hold, the work supplies a transparent reference case for MZ applied to linear systems, clarifying how unresolved variables enter through history-dependent terms. It identifies key ingredients for extensions to nonlinear systems, spectral filtering, and data-driven memory approximations, which may support reduced-order modeling in dynamical systems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and the recommendation to accept the manuscript. The referee summary correctly identifies the main contributions regarding the explicit Mori-Zwanzig formulation for partially observed linear systems.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives explicit closed-form Markovian, noise, and memory terms for linear time-invariant systems by expressing dynamics via the Koopman generator and orthogonal projections, then shows these recover the known reduced dynamics from the variation-of-constants formula. This equivalence is a direct consequence of linearity (explicit semigroup and linear evolution of the orthogonal complement) rather than a fitted parameter or self-referential definition. No self-citations are invoked as load-bearing premises, no ansatz is smuggled, and no uniqueness theorem from prior author work is used to force the result. The construction is presented as a transparent reference case that retains the Mori-Zwanzig operator structure while matching the classical formula, making the central claim independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract provides limited information on additional assumptions; the main ones are the linearity and the projection framework standard in the field.

axioms (2)
  • domain assumption The dynamical system is linear and time-invariant
    The derivation is developed specifically for this class of systems as stated in the abstract.
  • domain assumption The Koopman generator and projections onto resolved/unresolved components are applicable and well-defined
    Central to expressing the dynamics and deriving the contributions.

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Reference graph

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12 extracted references · 2 canonical work pages · 1 internal anchor

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