Neural Diffusion Intensity Models

• The paper introduces Neural Diffusion Intensity Models as diffusion-based latent-variable models for Cox processes, employing neural SDEs to capture stochastic event intensities.
• It replaces traditional MCMC methods with an amortized variational inference approach that simulates a drift-corrected posterior SDE for rapid and scalable estimation.
• Empirical results demonstrate that this method achieves similar accuracy to MCMC with significantly reduced computation time, making it practical for real-time continuous-time event analysis.

Searching arXiv for papers on neural diffusion intensity models and closely related diffusion-based intensity/density modeling. Neural Diffusion Intensity Models are diffusion-based latent-variable models for point process data in which the event-rate process itself is stochastic and evolves according to a neural SDE. In the formulation introduced for Cox processes, observed event times are generated by a latent stochastic intensity $Z_t$, while learning and posterior inference are performed with a variational family whose paths retain diffusion structure after conditioning on observations (Du et al., 27 Feb 2026). Within the broader diffusion literature, the phrase “intensity” is used in several distinct senses: as a point-process rate in Cox-process modeling, as total image intensity in conservation-constrained generative modeling, and as arithmetic intensity in systems work on diffusion LLMs. The point-process meaning is the technically specific sense associated with Neural Diffusion Intensity Models proper (Du et al., 27 Feb 2026).

1. Definition and scope

In the point-process setting, Neural Diffusion Intensity Models address Cox processes, also called doubly stochastic Poisson processes, where observed event times are driven by a latent stochastic intensity process $Z_t$ (Du et al., 27 Feb 2026). This formulation targets overdispersed point process data, meaning count variability exceeds what a standard Poisson model explains, and it treats inference over the latent intensity path as the central computational problem (Du et al., 27 Feb 2026).

The model class is diffusion-driven in a literal sense: the latent intensity is a diffusion solving an SDE with neural-network-parameterized drift. The corresponding marginal likelihood

$P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$

is intractable because the integral ranges over an entire path space of trajectories $Z$ (Du et al., 27 Feb 2026). The proposed framework replaces repeated MCMC-based posterior sampling with amortized variational inference built around a drift-corrected posterior SDE (Du et al., 27 Feb 2026).

The broader literature contains adjacent but non-identical uses of related terminology. “Discrete Spatial Diffusion” models exact conservation of total image intensity in discrete spatial domains, but its object is not an event intensity $\lambda(t\mid \mathcal H_t)$ or a Cox-process latent rate (Santos et al., 3 May 2025). “Diffusion Density Estimators” studies diffusion models as neural density estimators that compute log densities without solving a Probability Flow ODE; its target is likelihood evaluation for data samples rather than latent point-process intensity paths (Premkumar, 2024). “Diffusion-Augmented Neural Processes” explicitly states that it is not an intensity model in the usual point-process sense (Bonito et al., 2023). This terminological separation is important because the word “intensity” is overloaded across diffusion-model research.

2. Probabilistic formulation for Cox processes

The latent intensity $Z_t$ is modeled as a diffusion

$$dZ_t = b_\theta(Z_t,t)\,dt + \sigma(Z_t,t)\,dB_t,\qquad Z_0=z.$$

Here $b_\theta$ is the drift parameterized by a neural network, while $\sigma$ is the diffusion coefficient, typically fixed or chosen from a parametric form such as CIR (Du et al., 27 Feb 2026).

Conditional on the intensity path $Z$, the observed process $Z_t$0 is an inhomogeneous Poisson process with rate $Z_t$1. On $Z_t$2, the event likelihood is

$Z_t$3

so the joint model is

$Z_t$4

The resulting model can therefore be viewed as an infinite mixture of Poisson processes whose mixing distribution is induced by the SDE prior over intensities (Du et al., 27 Feb 2026).

This construction is aimed at nonparametric maximum likelihood over the marginal point-process distribution. The stated learning target is

$Z_t$5

but the path integral is not tractable in closed form (Du et al., 27 Feb 2026). That intractability motivates the variational diffusion formulation that gives the model class its name.

A plausible implication is that the model occupies the same conceptual niche for continuous-time event data that latent neural SDEs occupy for continuous observations, except that the observation model is Poissonian and the latent path is interpreted explicitly as an intensity.

3. Posterior diffusion structure via enlargement of filtrations

The central theoretical contribution is a posterior-structure theorem based on enlargement of filtrations. Conditioning on observed event times does not destroy the diffusion form of the latent intensity process; instead, the posterior remains a diffusion with the same diffusion coefficient and an additional drift correction (Du et al., 27 Feb 2026).

The variational or posterior family is written as

$Z_t$6

Theorem 2.1 gives the exact posterior SDE structure

$Z_t$7

with

$Z_t$8

The paper interprets this additional term as a score-like drift correction (Du et al., 27 Feb 2026).

Three structural consequences are explicit. The posterior remains a diffusion, the diffusion coefficient does not change, and the effect of conditioning is entirely absorbed into the drift through a conditional log-derivative term (Du et al., 27 Feb 2026). The paper notes that this is analogous in spirit to classifier guidance in diffusion models, where a prior trajectory is steered by a gradient of a log conditional density (Du et al., 27 Feb 2026).

This result matters because it makes the variational family exact in principle: the family used for inference is not merely heuristic but is structured to contain the true posterior under sufficient model capacity (Du et al., 27 Feb 2026). That alignment is the basis for the model’s maximum-likelihood interpretation.

4. Variational objective and amortized inference

Using the SDE-induced variational family $Z_t$9, the framework optimizes the usual ELBO

$P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$0

For the drift-corrected posterior-SDE family, the KL term has a closed form, yielding

$P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$1

The empirical objective is

$P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$2

Because the true posterior lies in the same SDE family in principle, the paper argues that with sufficient model capacity the variational gap vanishes and ELBO maximization coincides with maximum likelihood estimation (Du et al., 27 Feb 2026).

Posterior inference is amortized through an encoder that maps a variable-length sequence of event times into a drift correction. The encoder uses a Deep Sets-style permutation-invariant architecture,

$P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$3

where $P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$4 and $P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$5 are MLPs (Du et al., 27 Feb 2026). In the terminology used there, $P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$6 encodes per-event contributions, the sum aggregates event information after time $P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$7, and $P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$8 maps the aggregate together with current state and time to the drift correction (Du et al., 27 Feb 2026).

The practical significance is that posterior sample paths are obtained by directly simulating the corrected SDE in a single forward pass, rather than running fresh MCMC for every observation sequence (Du et al., 27 Feb 2026). This suggests an inference pipeline better suited to repeated or real-time posterior estimation than classical EM-plus-MCMC procedures.

5. Empirical behavior and reported performance

The synthetic experiments use a CIR latent intensity

$P_\theta(X)=\int P(X\mid Z)\,dP_\theta(Z)$9

with $Z$0 simulated Cox-process sequences on $Z$1, Euler–Maruyama discretization with $Z$2 steps, $Z$3 Brownian samples per observation for Monte Carlo gradients, and training for $Z$4 epochs with batch size $Z$5 and learning rate $Z$6 (Du et al., 27 Feb 2026). A time-inhomogeneous variant,

$Z$7

is also studied (Du et al., 27 Feb 2026).

On prior recovery, the learned drift $Z$8 qualitatively matches the true drift and reproduces sample statistics well (Du et al., 27 Feb 2026). Under an equal $Z$9-hour compute budget, the reported $\lambda(t\mid \mathcal H_t)$0 path deviations are summarized below.

True drift	EM loss	VI loss
$\lambda(t\mid \mathcal H_t)$1	6.159	5.792
$\lambda(t\mid \mathcal H_t)$2	6.721	4.989

Posterior inference quality is evaluated against high-fidelity MCMC. The amortized posterior tracks MCMC posterior sample paths closely for both complete data $\lambda(t\mid \mathcal H_t)$3 and partial observation $\lambda(t\mid \mathcal H_t)$4 (Du et al., 27 Feb 2026). The paper also studies amortization overfitting using a Wasserstein distance between amortized and MCMC posterior path measures, reporting that a train/test gap appears for small training sets and vanishes once training size exceeds about $\lambda(t\mid \mathcal H_t)$5 (Du et al., 27 Feb 2026).

The main computational claim concerns speed. For synthetic experiments, the variational method is typically $\lambda(t\mid \mathcal H_t)$6–$\lambda(t\mid \mathcal H_t)$7 orders of magnitude faster than MCMC-based posterior sampling while achieving similar predictive likelihoods (Du et al., 27 Feb 2026). Representative timings include full-horizon posterior inference on $\lambda(t\mid \mathcal H_t)$8: MCMC $\lambda(t\mid \mathcal H_t)$9m $Z_t$0s versus VI $Z_t$1s (Du et al., 27 Feb 2026). On a large U.S. bank call-center dataset with minute-level arrivals, $Z_t$2 Mondays for training, and observations heavily thinned with probability $Z_t$3 for stability, the learned mean intensity captures the main daily pattern, though the learned variance is less accurate (Du et al., 27 Feb 2026).

The weaker variance fit is attributed in the paper to fixing $Z_t$4 as $Z_t$5, and learning $Z_t$6 as another neural network $Z_t$7 is presented as a natural extension (Du et al., 27 Feb 2026).

6. Conceptual relations to adjacent diffusion research

Neural Diffusion Intensity Models for point processes belong to a wider family of diffusion-based probabilistic modeling methods, but neighboring approaches target different objects.

“Diffusion Density Estimators” examines diffusion models as neural density estimators and introduces a Monte Carlo path-integral estimator for log density that avoids the Probability Flow ODE (Premkumar, 2024). Its central transferable idea is to replace sequential ODE likelihood computation with a highly parallelizable Monte Carlo path integral built from analytic transition kernels (Premkumar, 2024). This is adjacent to NDIM in that both works seek tractable likelihood-related computation from diffusion machinery, but one addresses sample densities while the other addresses posterior inference over latent intensity paths.

“Discrete Spatial Diffusion: Intensity-Preserving Diffusion Modeling” uses the term “intensity” in an image-generation sense: pixel values are interpreted as discrete particles moving on a lattice under a continuous-time Markov jump process that preserves total intensity exactly in both forward and reverse processes (Santos et al., 3 May 2025). It therefore concerns conservation of mass or particle count, not a point-process event rate. The distinction is technical rather than terminological: its forward and reverse dynamics are particle-conserving jump processes on images, whereas NDIM models stochastic event intensities for Cox processes (Santos et al., 3 May 2025).

“Diffusion-Augmented Neural Processes” is diffusion-inspired conditional regression, but it explicitly states that there is no explicit point-process intensity function, event arrival rate, or temporal intensity modeling in the paper (Bonito et al., 2023). “Optical Diffusion Models for Image Generation” realizes a denoising diffusion model in passive optics and uses “output intensity” to refer to optical field intensity corresponding to the predicted noise term (Oguz et al., 2024). “Orchestrating Dual-Boundaries” uses “arithmetic intensity” to denote a hardware-performance quantity in diffusion LLM inference (Wei et al., 24 Nov 2025). These are all valid diffusion-related uses of the word, but they are separate from the point-process meaning formalized by NDIM.

A plausible implication is that the specific contribution of NDIM is not merely “using diffusion models for point processes,” but identifying a posterior family whose structure matches the exact conditioned path law of a diffusion-driven Cox process.

7. Limitations, extensions, and interpretation

The stated limitations are concrete. The diffusion coefficient $Z_t$8 is fixed, which may limit fit on real data; the encoder must approximate a complicated functional of future event times; training relies on Monte Carlo simulation and Euler discretization, introducing discretization bias; and experiments are mainly on one-dimensional point processes (Du et al., 27 Feb 2026).

The proposed extensions are likewise explicit: learning the diffusion coefficient $Z_t$9 with a neural network, extending to multivariate point processes, applying the method to marked or spatiotemporal point processes, using richer observation models beyond simple Cox or Poisson likelihoods, and extending the enlargement-of-filtrations approach to other continuous-time latent-variable models (Du et al., 27 Feb 2026).

The main interpretive claim is that posterior inference for diffusion-driven Cox processes can be reformulated as simulation of a drift-corrected SDE with unchanged diffusion coefficient (Du et al., 27 Feb 2026). In that sense, Neural Diffusion Intensity Models convert a path-space Bayesian inference problem into amortized simulation under a learned correction field. This suggests an overview of Cox-process statistics, neural SDE priors, and diffusion-style posterior guidance that is theoretically tied to maximum likelihood when the model family is sufficiently expressive (Du et al., 27 Feb 2026).