Diffusion-based Dynamical Systems

• Diffusion-based dynamical systems are mathematical frameworks characterized by deterministic, stochastic, and score-based diffusion mechanisms modeling transport and temporal evolution.
• They leverage rigorous methods such as hydrodynamic limits and CTRW theory to bridge microscale dynamics with macroscopic behavior in complex systems.
• Applications include turbulence modeling, uncertainty quantification, and control of nonlinear, high-dimensional systems across physical and network domains.

Diffusion-based dynamical systems are mathematical and computational frameworks in which key aspects of transport, uncertainty, and temporal evolution are driven or modeled by diffusion mechanisms. These mechanisms include deterministic diffusion (arising from system structure, conservation laws, or coupling), stochastic diffusion (arising from noise or probabilistic modeling), and modern score-based generative processes (where diffusion models are trained to represent dynamical laws or probabilistic flow maps). The concept encompasses both the emergence of macroscopic diffusion from deterministic microscale dynamics and the explicit construction of generative models for dynamical systems using diffusion-based methods.

1. Deterministic and Stochastic Diffusive Dynamics

Deterministic lattice dynamical systems with local conservation laws and weak coupling produce macroscopic diffusive behavior under suitable scaling. When each lattice site possesses a slow, conserved variable (e.g., energy) and chaotic fast degrees of freedom, weak coupling drives the slow variables toward a hydrodynamic regime governed by the heat equation: $\partial_t e(t,x) = D\,\Delta e(t,x)$ with diffusion constant D given by a Green-Kubo-type formula involving the microscopic currents. Rigorous results demonstrate this emergence of diffusion in both discrete-time, weakly coupled hyperbolic maps and coupled map lattices, provided the fast subsystem admits a unique SRB measure with exponential mixing and the coupling is sufficiently weak (Kupiainen, 2010, Bricmont et al., 2011).

Stochastic diffusion arises when deterministic systems are subjected to random perturbations or when stochastic reaction networks are considered. In the context of dissipative dynamics with many coexisting attractors, bounded random perturbation leads to hopping (random transitions) among basins, and the system is described as a random dynamical system with holes. Transport among basins becomes anomalous when hyperbolicity is weak, yielding subdiffusive scaling of the mean-square displacement and power-law escape statistics (Rodrigues et al., 2014).

2. Mathematical Formulations and Anomalous Diffusion

A general randomly perturbed dissipative system may be formulated as

$$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$

with $f$ a $C^1$ diffeomorphism and bounded random kicks $\varepsilon_j$. Each attractor $\Lambda_i$ has an associated basin $W^s(\Lambda_i)$. As the amplitude $\xi$ increases past a critical value, orbits may escape basins through holes of positive measure, resulting in intermittent "hopping" transport.

In hyperbolic regimes (e.g., strong dissipation), escape times from basins are exponentially distributed: $P(t) \sim a\,e^{-\alpha t}$ yielding normal or fast basin mixing. In near–non-hyperbolic regimes (e.g., weak dissipation), trapping near marginally stable structures causes heavy-tailed statistics: $P(t) \sim b\,t^{-\beta}, \quad 1<\beta<3$ Power-law escape times underpin subdiffusive transport along the diffusive coordinate $$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$0: $$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$1 with position distributions exhibiting stretched-exponential, non-Gaussian tails

$$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$2

Anomalous, non-Brownian diffusion emerges because the bounded noise both washes out fine structure and reinforces metastability near sticky regions (Rodrigues et al., 2014).

Continuous-Time Random Walk (CTRW) theory quantitatively reproduces these effects via a waiting-time distribution $$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$3 and a narrow jump-length distribution. The resulting dynamics, encoded in the Montroll–Weiss equation, predict the same MSD exponents and position PDF tails as observed numerically.

3. Network-level and Hierarchical Diffusion

Diffusion processes on networks exhibit multiscale structure and may be described by Markov chains with aggregated variables (modules or communities). Weak inter-module coupling leads to a clear separation of time scales: fast equilibration within modules and slow diffusion between them. The reduced module dynamics satisfy transport equations analogous to Fick's law, with effective diffusion constants and possible directional flux (driven by asymmetries in network structure, fitness, or coupling): $$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$4 Entropy production can be decomposed into macroscopic inter-module and microscopic intra-module terms. In hierarchical (nested) networks, iteration of the time-scale separation yields a hierarchy of diffusion times; each corresponds to a gap in the eigenspectrum of the random-walk operator, enabling the discovery of hidden structural levels (Siudem et al., 2013).

4. Score-based Diffusion Models as Dynamical Operators

Score-based diffusion models provide a flexible, probabilistic framework for modeling solution operators, dynamics, and inference in high-dimensional dynamical systems. In recent advances, dynamical evolution is cast as a generative process, where the forward SDE progressively perturbs a sample from the target distribution (e.g., solutions, closure terms) to noise, and the learned reverse process reconstructs samples conditioned on system parameters, observed states, or partial information (Haitsiukevich et al., 2024, Dong et al., 25 Jun 2025, Yang et al., 2 Feb 2026).

Conditional diffusion models are trained with denoising score-matching, often within a latent space constructed by autoencoders. Examples include:

• Latent score-based conditional modeling for non-local, stochastic closure in nonlinear PDEs, enabling efficient sampling of subgrid-scale fluxes (Dong et al., 25 Jun 2025)
• Training-free score-based generative maps for parameter-dependent SDEs, using joint kernel-weighted Monte Carlo for the conditional score (Yang et al., 2 Feb 2026)
• Diffusion models acting as neural operators for forward and inverse PDE problems, partial observability, and uncertainty quantification (Haitsiukevich et al., 2024)

The flexibility of these models allows unified treatment of various tasks: forecasting, imputation, inverse problems, uncertainty propagation, and event-constrained sampling. Statistical properties such as long-range dependencies, tail events, and multimodality are naturally represented.

5. Dynamical Regimes and Phase Transitions in Discrete and Continuous Diffusion

In score-based (especially discrete) diffusion models of dynamical systems, the generation process exhibits distinct dynamical regimes:

• Regime I (noise-dominated): Reverse trajectories exhibit high diversity, with class or global structure not yet emerging.
• Regime II (speciation): At a critical noise threshold ($$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$5), global structure appears (e.g., class magnetization bifurcates in Ising models); diversity remains high.
• Regime III (collapse): Beyond a "condensation" transition ($$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$6), trajectories concentrate onto specific class exemplars, diversity collapses (Takahashi et al., 13 Apr 2026).

These transitions correspond, respectively, to a second-order (high-temperature) instability (speciation) and a condensation phenomenon in the Random Energy Model (collapse). Analytical criteria for $$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$7 and $$x_{j+1} = f(x_j) + \varepsilon_j, \quad \|\varepsilon_j\| < \xi$$8 can be formulated in terms of the data covariance spectrum and entropy, and their scaling mirrors that found in continuous-Gaussian models.

6. Practical Applications and Computational Aspects

Diffusion-based dynamical systems methods are applied to:

• Prediction and closure in turbulence, meteorology, and complex multiscale flows, where stochasticity and non-locality are essential (Dong et al., 25 Jun 2025)
• Probabilistic neural operators and data assimilation, where conditional sampling, joint learning of forward/inverse tasks, and uncertainty quantification are critical (Haitsiukevich et al., 2024, Chakraborty et al., 13 Mar 2026, Finzi et al., 2023)
• Surrogate modeling of high-dimensional, chaotic systems using graph-based diffusion architectures for unstructured data, adaptive sensor placement, and likelihood-guided posterior sampling (Chakraborty et al., 13 Mar 2026)
• Control of nonlinear dynamical systems via diffusion maps into the space of feedback policies, providing a principled generative-sampling perspective for steering state distributions (Elamvazhuthi et al., 2024)
• Network diffusion control with feedback, reinforcement learning, and targeted topology design (Chan et al., 2015), as well as analytical quantification of diffusive capacity in complex and multiplex networks (Schieber et al., 2021)

Table: Examples of Diffusion-based Dynamical System Mechanisms

Setting	Key Mechanism	arXiv ID
Lattice dynamics	Diffusive hydrodynamic limit from weakly coupled hyperbolic maps	(Kupiainen, 2010)
Reaction-diffusion	Diffusion-independence at steady state for complex balanced networks	(Vu et al., 2018)
Stochastic PDEs	Conditional score-based generative modeling (latent, joint kernel)	(Dong et al., 25 Jun 2025, Yang et al., 2 Feb 2026)
Network transport	Coarse-grained Fickian flux and entropy production via adiabatic approx	(Siudem et al., 2013)
Dynamical inference	Score-based posterior sampling, event constraint satisfaction	(Finzi et al., 2023)

7. Fundamental Theoretical Results and Limitations

Key theoretical contributions include:

• Hydrodynamic limit theorems for weakly coupled deterministic systems, with explicit construction of the diffusion constant from microscopic dynamics (Kupiainen, 2010, Bricmont et al., 2011)
• Characterization of conditions where spatial diffusion can be neglected in networked reaction systems (complex balance, linearity) (Vu et al., 2018)
• Complete statistical correspondence of numerically observed subdiffusive transport and position distributions to predictions from stochastic CTRW theory (Rodrigues et al., 2014)
• Asymptotic normality and provable convergence of event-conditioned sampling in diffusion models as noise vanishes (Finzi et al., 2023, Takahashi et al., 13 Apr 2026)

Limitations include the need for exponential mixing (strong chaos) for emergent diffusion, difficulties in representing ultra-rare tail events in certain inference schemes, and, in high-dimensional or complex-geometry problems, the need for computationally efficient surrogate architectures and approximations to the score and filtering operators.

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References:

• "Diffusion in randomly perturbed dissipative dynamics" (Rodrigues et al., 2014)
• "Origins of Diffusion" (Kupiainen, 2010)
• "Diffusion-dynamics laws in stochastic reaction networks" (Vu et al., 2018)
• "Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models" (Dong et al., 25 Jun 2025)
• "Dynamical Regimes of Discrete Diffusion Models" (Takahashi et al., 13 Apr 2026)
• "Characterization and Control of Diffusion Processes in Multi-Agent Networks" (Chan et al., 2015)
• "Diffusion maps kernel ridge regression" (Song et al., 19 Dec 2025)
• "Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systems" (Haitsiukevich et al., 2024)
• "Network diffusion capacity unveiled by dynamical paths" (Schieber et al., 2021)
• "Training-free score-based diffusion for parameter-dependent stochastic dynamical systems" (Yang et al., 2 Feb 2026)
• "Denoising Diffusion-Based Control of Nonlinear Systems" (Elamvazhuthi et al., 2024)
• "Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems" (Chakraborty et al., 13 Mar 2026)
• "User-defined Event Sampling and Uncertainty Quantification in Diffusion Models for Physical Dynamical Systems" (Finzi et al., 2023)
• "A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results" (Gidea et al., 2014)
• "Diffusion on hierarchical systems of weakly-coupled networks" (Siudem et al., 2013)