Mori–Zwanzig: Reduced Order Modeling

• Mori–Zwanzig formalism is a rigorous operator-based framework that derives reduced models by capturing unresolved variables through non-Markovian memory kernels and stochastic forces.
• The method uses projection operators to split dynamics into instantaneous Markovian terms and memory integrals, effectively encoding the back-reaction of eliminated variables.
• It has broad applications in turbulence, climate dynamics, molecular simulations, and quantum systems, with modern data-driven techniques improving its practical implementation.

The Mori–Zwanzig (MZ) formalism is a mathematically rigorous operator-based framework for deriving reduced-order models of high-dimensional dynamical systems, explicitly capturing the influence of unresolved (or "eliminated") variables through non-Markovian memory kernels and fluctuating forces. Originally developed in nonequilibrium statistical mechanics, the formalism provides the foundation for the Generalized Langevin Equation (GLE), widely used for coarse-graining, model reduction, and uncertainty quantification in disciplines ranging from turbulence and climate dynamics to molecular dynamics and quantum systems.

1. Operator Framework: Projection, Memory, and Fluctuating Forces

The central object in the Mori–Zwanzig formalism is an exact evolution equation for a set of selected ("resolved") observables or variables in a high-dimensional system. Suppose the full system evolves as $dx/dt = Lx$ or, in more generality, $dx/dt = f(x)$, and let $P$ denote a linear projection operator onto the resolved subspace, with $Q = I-P$. The formalism yields

$$\frac{d}{dt} [P x(t)] = P L x(t) + \int_0^t K(t-s)\,[P x(s)]\,ds + f(t),$$

with

$$K(\tau) = P L\,e^{\tau Q L} Q L P, \qquad f(t) = e^{t Q L} Q L x(0).$$

The first term $P L x(t)$ is the Markovian (instantaneous, memoryless) contribution; the second is a non-local memory integral that accounts for the back-reaction of unresolved variables; and $f(t)$ is the (orthogonal) noise, generated from the unresolved initial condition and orthogonal evolution (Tian et al., 2021, Parish et al., 2017, Woodward et al., 2023, 1803.02826, Vrugt, 2021).

Physically, $P L x(t)$ describes how the resolved variables would evolve if the unresolved ones were slaved instantaneously; the memory kernel $K$ weights the influence of past resolved states mediated by unresolved dynamics; and the noise term encodes the stochastic component due to unresolved initial conditions.

2. Variants of Projections and Equivalence with Optimal Prediction

Projection choices are central in the MZ framework. Standard options include:

• Finite-rank linear projections: project deterministic quantities onto a basis (e.g., velocity moments, Legendre polynomials, finite elements).
• Conditional expectation projections: $dx/dt = f(x)$0 (Zwanzig projection).

For quantum or statistical systems, a Hilbert-space inner product such as $dx/dt = f(x)$1 or time-ensemble averaging is used to construct orthogonal projections. Nonlinear projections (Zwanzig projectors) can be used to capture additional structure (e.g., through conditional expectation over manifolds) (Jung et al., 2023).

The MZ formalism is closely related to the optimal prediction framework, in which the trajectory of the unresolved variables is replaced by its conditional expectation, leading to the same GLE structure (Parish et al., 2016).

3. Memory Kernel and Orthogonal Dynamics

The non-Markovian memory kernel is formally

$dx/dt = f(x)$2

which acts as an operator-valued convolution on the history of the resolved variables. Its derivation relies on a Dyson or Duhamel formula splitting the evolution operator $dx/dt = f(x)$3 into projected and orthogonal components (Tian et al., 2021, Parish et al., 2017, Vrugt, 2021).

The noise (fluctuating force) term $dx/dt = f(x)$4 is governed by the orthogonal dynamics equation, evolving in the Q-subspace: $dx/dt = f(x)$5 This term is typically orthogonal (in the $dx/dt = f(x)$6 sense or equivalent inner product) to all resolved observables at all times. In the context of stochastic processes, fluctuation–dissipation relationships can be derived linking the autocorrelation of $dx/dt = f(x)$7 to the memory kernel $dx/dt = f(x)$8 (Tian et al., 2021, 1803.02826, Vrugt, 2021, Jung et al., 2023, Woodward et al., 2023).

4. Data-Driven and Algorithmic Extraction of MZ Operators

Modern developments replace analytical evaluation of $dx/dt = f(x)$9 and $P$0 by data-driven procedures:

• Algorithmic extraction (linear basis): For time series of resolved observables $P$1, compute lagged covariances $P$2; recursively solve for Markov and memory matrices, and extract orthogonal noise samples (Tian et al., 2021, Lin et al., 2021, Woodward et al., 2023).
• Convex regression: In a finite-dimensional projection, GLE coefficients are learned as a closed-form regression problem by solving for $P$3 in

$P$4

with $P$5, using observed statistics.

• Generalized Fluctuation–Dissipation verification: The anti-self-adjointness of the Liouville operator enables matrix-level consistency checks between the learned memory kernel and the statistics of noise samples (Tian et al., 2021, Lin et al., 2021).

For nonlinear or partially observable systems, MZ kernels can be learned via extensions such as regression in extended dynamic mode decomposition (EDMD) bases or deep neural networks ("Deep MZ") (Wang et al., 22 Jun 2026, Lin et al., 2021).

5. Practical Applications: Multiscale Modeling, Turbulence, Model Reduction

The MZ formalism is adapted across continuum and discrete domains:

• Turbulence and Large Eddy Simulation (LES): The formalism provides a foundation for non-Markovian subgrid closures, where the memory kernel naturally represents backscatter and nonlocal interactions. In LES, MZ-based closures outperform ad hoc Markovian models and even capture features such as finite memory, energy decay, subgrid transfers, and nonlocal stabilization (Parish et al., 2016, Parish et al., 2017, Tian et al., 2021, Parish et al., 2016).
• Uncertainty Quantification: The MZ approach reduces the cost of solving systems induced by polynomial chaos expansions by constructing reduced models for select chaotic modes, transforming memory integrals into auxiliary ODE hierarchies and enabling adaptive Markovian approximation (Stinis, 2012, 1803.02826).
• Delay and Distributed-Memory Models: MZ projection yields systematic delay or distributed-delay equations (e.g., for climate dynamics), with explicit delay and non-Markovian terms stemming from physical propagation times or feedbacks (Falkena et al., 2019).

In each instance, the form of the closure is imposed by the mathematical structure of the projection and is parameter-free when the memory timescale or truncation is determined dynamically (e.g., Germano identity for LES) (Parish et al., 2016).

6. Theoretical Extensions: Nonlinear Projection, Time-Dependent Hamiltonians, Deep Learning

• Nonlinear projection operators: When used in coarse-graining with nonlinear dynamics (e.g., anharmonic traps or non-Gaussian observables), nonlinear Zwanzig projectors properly capture higher-order corrections missed by linear projections. This modification of the memory kernel is essential for accurate statistics in, e.g., stochastic generalized Langevin equations (Jung et al., 2023).
• Time-dependent Hamiltonians: The standard MZ formalism extends to systems with explicit time dependence via time-ordered exponentials, yielding integro-differential equations with two-time memory kernels and time-dependent projection operators. Markovian approximations in this context lead to irreversible transport equations with explicit time-dependence in the transport coefficients (e.g., relevant to NMR in varying fields) (Vrugt et al., 2019).
• Deep learning architectures: The layer-wise evolution of deep networks can be reinterpreted as a discrete MZ system, where memory operators express "effective depth" and elucidate why deep networks with vanishing residual memory act as shallow parameterizations. Under contraction mappings, memory terms decay exponentially with depth, enabling rigorous reduction of depth or width of networks while preserving expressiveness (Venturi et al., 2022).

7. Interpretation, Irreversibility, and Physical Significance

The decomposition in the MZ formalism provides a precise account of emergent irreversibility, the origin of stochasticity and noise in macroscopic dynamics, and the systematic connection between reversible microdynamics and irreversible transport (e.g., Green–Kubo relations, H-theorem). The validity of Markovian closures and effective autonomous macrodynamics can be rigorously justified via decay properties of the memory kernel (autonomy arises when slow variables are properly chosen and memory is short) (Vrugt, 2021).

The framework also clarifies subtleties about "epistemic vs. ontic" probabilities: the MZ approach uses both the full microstate density and a "relevant" maximum-entropy density for projections, thereby connecting dynamical and information-theoretic perspectives in statistical mechanics (Vrugt, 2021).

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References

• (Tian et al., 2021) Data Driven Learning of Mori-Zwanzig Operators for Isotropic Turbulence
• (Parish et al., 2017) A Unified Framework for Multiscale Modeling using the Mori-Zwanzig Formalism and the Variational Multiscale Method
• (Parish et al., 2016) Non-Markovian Closure Models for Large Eddy Simulations using the Mori-Zwanzig Formalism
• (Woodward et al., 2023) Mori-Zwanzig mode decomposition: Comparison with time-delay embeddings
• (Lin et al., 2021) Data-driven learning for the Mori-Zwanzig formalism: a generalization of the Koopman learning framework
• (Wang et al., 22 Jun 2026) Partial Observation of Linear Systems with the Mori-Zwanzig Formalism
• (Parish et al., 2016) A Dynamic Subgrid Scale Model for Large Eddy Simulations Based on the Mori-Zwanzig Formalism
• (1803.02826) Mori-Zwanzig reduced models for uncertainty quantification
• (Stinis, 2012) Mori-Zwanzig reduced models for uncertainty quantification I: Parametric uncertainty
• (Falkena et al., 2019) Derivation of Delay Equation Climate Models Using the Mori-Zwanzig Formalism
• (Venturi et al., 2022) The Mori-Zwanzig formulation of deep learning
• (Jung et al., 2023) Dynamic Coarse-Graining of Linear and Non-Linear Systems: Mori-Zwanzig Formalism and Beyond
• (Vrugt, 2021) Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism
• (Vrugt et al., 2019) Mori-Zwanzig projection operator formalism for systems with time-dependent Hamiltonians
• (Parish et al., 2017, Tian et al., 2021, Lin et al., 2021, Woodward et al., 2023, Parish et al., 2016, Parish et al., 2016)