Non-Identical Diffusion Model Insights

• Non-identical diffusion models are heterogeneous processes where noise or transport properties vary across dimensions, scales, or channels.
• They implement distinct local versus global noise control and geometry-aware transport by using element-wise time indicators and non-uniform increments.
• These models generalize traditional uniform diffusion, enhancing robustness and adaptability in generative modeling, physical transport, and multi-channel systems.

Taken together, the literature suggests that a non-identical diffusion model is a diffusion process in which the forward or reverse dynamics are not shared uniformly across all coordinates, states, scales, channels, or interacting subsystems. In generative modeling, this includes element-wise time indicators, non-uniform schedules, constrained-domain mirror dynamics, non-Gaussian increment laws, and learnable spatial stochastic differential equations; in physical transport, it includes state-dependent diffusivity, random diffusivity, distributed-order diffusion, multi-path diffusion, and cross-diffusion among interacting species (Yang et al., 1 Sep 2025, Tae, 2023, Tupper et al., 2012).

1. Conceptual definition

The standard diffusion-model template uses a single scalar time or noise index and a shared noise law across dimensions. In discrete DDPM-style notation, the forward chain is written as

$$q(x_0,\dots,x_T)=q_d(x_0)\prod_{t=1}^T T_{q_n}(x_t\mid x_{t-1};\beta_t),$$

where $T_{q_n}$ depends on a chosen noise family $q_n$ and a time-dependent diffusion strength $\beta_t$ (Marjieh et al., 2022). In state-space form, a standard score-based forward process is

$d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$

with reverse-time dynamics determined by the score $\nabla_x \log p_t(x)$ (Tae, 2023).

A non-identical diffusion model relaxes this uniformity. In the machine-learning papers considered here, non-identicality appears when different coordinates, scales, or trajectories are assigned different noise progressions, different geometries, or different increment distributions. In the transport literature, it appears when diffusion coefficients vary across space, across species, or across time, or when multiple diffusion channels are coupled. This suggests that “non-identical” refers less to a single formalism than to a family of constructions in which diffusion is heterogeneous rather than globally shared.

Two recurring contrasts organize the topic. The first is global versus local noise control: standard generative diffusion uses a scalar-valued time index, whereas non-identical formulations may use an element-wise time indicator or a scale-dependent schedule (Yang et al., 1 Sep 2025, Batzolis et al., 2022). The second is fixed versus geometry-aware transport: standard models typically assume isotropic Gaussian increments in unconstrained $\mathbb{R}^d$, whereas non-identical variants may adapt noise to a simplex, a bounded domain, a learned Riemannian metric, or a state-dependent diffusion coefficient (Tae, 2023, Du et al., 2022, Tupper et al., 2012).

2. Element-wise non-identical diffusion in generative modeling

The most explicit use of the term appears in wireless channel generation, where a “non-identical diffusion model” replaces the scalar time index with an element-wise time indicator so that different entries of the input can follow different noise progressions (Yang et al., 1 Sep 2025). The motivation is MIMO-OFDM channel recovery: pilot schemes produce initial channel estimates with highly uneven reliability across antennas and subcarriers, so a single global noise level is an inadequate description of the initialization.

In that formulation, the forward process is specified by a vector-valued schedule $\bm{\alpha}_t \in (0,1]^d$, with element-wise monotonicity and boundary conditions

$$\bm{\alpha}_0=\bm{1},\qquad \lim_{t\to T}\bm{\alpha}_t=\bm{0}.$$

The forward Itô SDE is

$$d \bm{H}_t = [\log \bm{\alpha}_t]'\circ \bm{H}_t\, dt + \sqrt{-2[\log \bm{\alpha}_t]'}\circ d\bm{W}_t,$$

and its marginal law has the closed form

$T_{q_n}$0

with $T_{q_n}$1 (Yang et al., 1 Sep 2025). The corresponding non-identical DDIM ODE uses the conditional expectation

$T_{q_n}$2

and is proved correct in the sense that, if initialized from the appropriate forward marginal, it recovers the data distribution at terminal time (Yang et al., 1 Sep 2025).

For MIMO-OFDM, the paper introduces a matrix matching the input size to control element-wise noise progression. The initial estimate $T_{q_n}$3 is combined with a reliability map $T_{q_n}$4, leading to

$T_{q_n}$5

so that reliable elements start closer to the clean end of the trajectory and unreliable ones start deeper in noise (Yang et al., 1 Sep 2025). Because the backbone network is an interleaved MLP-Mixer, the time indicator is not embedded as a scalar but via dimension-wise averaging and antenna/subcarrier-specific embeddings. The paper evaluates row-wise, column-wise, and joint embedding schemes, together with uniform, waterfilling, and hybrid time-stepping rules, and reports that $T_{q_n}$6-waterfilling gives the best final NMSE in most settings (Yang et al., 1 Sep 2025).

A central implication is that identical diffusion remains a special case. If all coordinates share the same schedule, $T_{q_n}$7, the construction collapses to standard identical diffusion. Non-identical diffusion therefore generalizes the scalar-time formulation rather than replacing it.

3. Other machine-learning realizations

Several other generative-model papers instantiate the same heterogeneity principle through different mechanisms. In “Non-Uniform Diffusion Models,” non-uniform diffusion leads to multi-scale diffusion models with similar structure to multi-scale normalizing flows, achieves a better FID score than the standard uniform diffusion model in the same or less training time, and generates samples $T_{q_n}$8 times faster at $T_{q_n}$9 resolution; the paper also reports that non-uniform diffusion yields a novel estimator for the conditional score function with on-par performance relative to the state-of-the-art conditional denoising estimator (Batzolis et al., 2022). Here, non-identicality is scale-dependent: different resolutions effectively follow different diffusion schedules.

In “Mirror Diffusion Models,” the departure from identicality is geometric rather than multi-scale. Diffusion is adapted to constrained domains such as the simplex by moving to a dual Euclidean space through a mirror map $q_n$0, running diffusion there, and mapping back through $q_n$1 (Tae, 2023). For simplex diffusion, the negative entropy mirror map

$q_n$2

induces the inverse mirror map

$q_n$3

and the reverse SDE contains the Hessian factor $q_n$4, so the noise geometry is no longer isotropic in primal coordinates (Tae, 2023). In this sense, the forward kernel is non-identical because it is shaped by the constrained domain.

“Non-Normal Diffusion Models” relaxes a different assumption: the default Gaussian law for the diffusion increments $q_n$5 (Li, 2024). The main theoretical statement is that, when the step size goes to zero, the reversed process is invariant to the distribution of these increments, provided the increment law matches the required first and second moments. This expands the admissible design space from Gaussian increments to a class that includes Laplace and Uniform choices, and correspondingly changes the per-step training loss from the standard quadratic form to alternative $q_n$6-like or hybrid objectives (Li, 2024). The paper reports competitive density-estimation and generative-modeling performance on standard image datasets and shows that different choices of the increment distribution produce qualitatively different generated samples (Li, 2024).

“A Flexible Diffusion Model” moves the same idea into the forward SDE itself. It parameterizes the spatial part of the SDE through a symmetric positive-definite metric field $q_n$7 and an antisymmetric matrix $q_n$8, yielding a class of forward processes that preserve a Gaussian stationary distribution and recover VP, VE, sub-VP, and critically-damped Langevin as special cases (Du et al., 2022). The forward geometry is therefore not fixed in advance; it is learned within a constrained family.

A final variant, “Non-Cross Diffusion,” addresses semantic inconsistency in ODE-based generative flows by augmenting the input dimension with a self-conditioned predicted noise term $q_n$9, so that different flows are distinguished in an extended state space and do not cross there (Zheng et al., 2023). The paper reports a substantial reduction in semantic inconsistencies at different inference steps and improved diffusion-model performance. This is another form of non-identicality: the reverse dynamics depend on $\beta_t$0, not only on $\beta_t$1 (Zheng et al., 2023).

4. State-dependent, random, and distributed-order diffusion in transport theory

Outside generative modeling, non-identical diffusion has a long history in stochastic transport. One canonical example is state-dependent diffusivity. “A Paradox of State-Dependent Diffusion and How to Resolve It” considers a domain split into two equal regions with different diffusion coefficients and shows that specifying the diffusion rate as a function of position is not enough to characterize the behaviour of a system, even in the absence of external forces (Tupper et al., 2012). The paper resolves the ambiguity by proposing a framework in which both the diffusion coefficient $\beta_t$2 and the equilibrium probability density $\beta_t$3 are specified, with drift

$\beta_t$4

This directly addresses a common misconception: a non-identical local diffusion coefficient does not by itself determine long-run occupancy.

A different class consists of random-diffusivity models. In “Diffusing diffusivity,” the diffusivity $\beta_t$5 itself performs a stochastic evolution, producing linear mean-squared displacement together with a non-Gaussian displacement distribution (Chubynsky et al., 2014). For an exponential diffusivity distribution

$\beta_t$6

the short-time displacement density becomes exactly exponential,

$\beta_t$7

even though the MSD remains $\beta_t$8 (Chubynsky et al., 2014). The paper explicitly describes this as “diffusivity memory” without direction memory. “Random diffusivity from stochastic equations” broadens this idea by using a generalized Gamma law for the diffusivity and comparing a static-random-diffusivity model with a dynamic diffusing-diffusivity model; the former preserves non-Gaussianity at all times, while the latter crosses over from non-Gaussian to Gaussian displacement statistics at long times (Sposini et al., 2018).

A third strand is distributed-order diffusion. “Distributed order Hausdorff derivative diffusion model to characterize non-Fickian diffusion in porous media” introduces

$\beta_t$9

which is equivalent to a diffusion equation with a nonlinear time-dependent diffusion coefficient (Liang et al., 2018). For the weight $d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$0, the effective diffusivity is derived explicitly, and the mean-squared displacement becomes proportional to a dilogarithm. The paper states that the model can capture accelerating and decelerating anomalous diffusion as well as ultraslow diffusion by varying $d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$1, and reports that tracer transport in water-filled pore spaces of two-dimensional Euclidean media is well described with $d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$2, whereas pore-solid prefractal porous media are accurately fit by the single-order Hausdorff model (Liang et al., 2018). Here non-identicality appears as a continuum of diffusion orders rather than a single exponent.

5. Multi-channel, multi-species, and nonlocal heterogeneous systems

Another major family of non-identical diffusion models arises when several transport channels or species coexist. In “Double diffusivity model under stochastic forcing,” the core model consists of two coupled Fick-type equations,

$d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$3

representing regular and high-diffusivity paths with exchange between them (Chattopadhyay et al., 2017). Eliminating the two fields yields a higher-order internal-length-gradient equation with inertia, viscous, and biharmonic terms, and the paper argues that adding stochastic forcing supplies a missing link between deterministic theory and experimentally verified relaxation times in nanopolycrystals (Chattopadhyay et al., 2017). Non-identicality here is channel-wise: the same species diffuses through paths with distinct diffusivities $d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$4 and $d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$5.

For interacting species, “Diffusion of multiple species with excluded-volume effects” derives a continuum cross-diffusion system from finite-size hard-core Brownian particles (Bruna et al., 2012). The resulting PDE shows that excluded-volume interactions among particles of the same species enhance the effective diffusion rate, whereas particles of the other species diminish it. The paper emphasizes that this resolves two notions often confounded in the literature, namely collective diffusion and self-diffusion, and derives explicit expressions for both in dilute regimes (Bruna et al., 2012).

A related ecological formulation appears in “Competitive-cooperative models with various diffusion strategies,” where dispersal is governed by

$d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$6

with $d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$7 representing a chosen dispersal strategy (Braverman et al., 2016). The paper shows that, if one species chooses $d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$8 proportional to the carrying capacity and the other does not, the first drives the second to extinction; if the ideal free distribution is attained as a combination of the two strategies, then there is a unique globally stable coexistence equilibrium; and if both species choose the same non-optimal strategy, higher diffusion rates are negative while higher intrinsic growth rates are positive for survival (Braverman et al., 2016). The diffusion operator is therefore non-identical in both structure and intensity.

Nonlocal heterogeneous diffusion extends the same principle to jump processes. “On the modelling of spatially heterogeneous nonlocal diffusion” defines

$d x = \mathbf{f}(x,t)\,dt + g(t)\,d\mathbf{w},$9

and introduces “deciding factors” that single out a nonlocal diffusion model, typically the total jump rate and the average jump length (Alfaro et al., 2021). In the singular limit of focusing kernels, different deciding factors recover different local heterogeneous diffusion laws, including Chapman-type, Fick-type, and Wereide-type forms. The paper also shows that steady-state profiles depend on these deciding factors, yielding preferential positions of individuals (Alfaro et al., 2021).

6. Theoretical themes, robustness, and recurring cautions

Across these literatures, several theoretical themes recur. One is that non-identicality often enlarges the model class without necessarily destroying the target stationary or terminal law. In the generative setting, non-normal increments with matched first and second moments converge to the same limiting reverse process as Gaussian increments when the step size goes to zero (Li, 2024). In the element-wise-time formulation, vector-valued schedules retain the correctness of DDIM-style generation (Yang et al., 1 Sep 2025). In the flexible-SDE framework, learned $\nabla_x \log p_t(x)$0 and $\nabla_x \log p_t(x)$1 preserve a Gaussian stationary distribution by construction (Du et al., 2022).

A second theme is that heterogeneity and robustness are not the same as arbitrariness. “Analyzing Diffusion as Serial Reproduction” shows that the sampling process of an optimal diffusion model approximates the true data distribution irrespective of the choice of noise family, so long as the noise schedule $\nabla_x \log p_t(x)$2 is chosen such that it alters the input distribution gradually (Marjieh et al., 2022). The same paper therefore distinguishes weak sensitivity to the noise family from strong sensitivity to scheduling: heterogeneous kernels are compatible with good generation, but abrupt schedules make the stepwise Bayesian inversion hard.

A third theme is that a non-identical diffusion coefficient or schedule does not by itself determine all emergent behavior. In state-dependent physical diffusion, $\nabla_x \log p_t(x)$3 alone does not determine equilibrium occupancy (Tupper et al., 2012). In generative modeling for MIMO-OFDM, an element-wise time matrix alone is not enough; performance depends materially on how that matrix is embedded, how it is stepped through time, and which reliability patterns are seen during training (Yang et al., 1 Sep 2025). In nonlocal ecological diffusion, the diffusivity matrix alone does not characterize the local limit when jump rate and jump length are decided at different points (Alfaro et al., 2021).

A final point concerns terminology. A plausible interpretation is that “non-identical diffusion model” has become a cross-domain label for diffusion processes that replace globally shared transition laws with heterogeneous ones. In machine learning, this usually means heterogeneity across coordinates, scales, trajectories, domains, or increment laws; in transport theory, it usually means heterogeneity across state, channel, time, or species. What unifies these uses is not a single canonical equation but a common structural move: diffusion is made local, selective, or geometry-dependent rather than identical everywhere.