Diffusion Processes

The inverse problem of reconstructing optical properties, specifically absorption and scattering coefficients, in layered biological media from time-domain reflectance measurements remains a significant challenge for traditional analytical models. Inverse solvers based on the diffusion equation often struggle with structural heterogeneity, frequently yielding poor accuracy for superficial absorption and deep-layers scattering. In this work, we propose a machine learning framework as an alternative approach to reconstruct the optical properties of a bilayered medium, benchmarking its efficiency and accuracy against model-based algorithms. To overcome the intrinsic approximations of diffusion theory and inverse reconstruction, we generated a robust synthetic dataset of forward DTOF using exact Monte Carlo simulations at multiple source-detector distances. A machine learning pipeline was then trained on this dataset and validated against state-of-the-art model-based reconstruction methods. Besides the significant reconstruction speed-up, the machine learning approach achieves higher accuracy than model-based inverse solvers, further providing an estimate of the parameter space dimensionality without requiring any a priori information about the number of layers in the investigated geometry. Further enhancements in the reconstruction accuracy can be expected in future extensions of this work, by training the pipeline over multiple DTOF curves from the same medium, in a joint multi-distance reconstruction approach.

Natural load-bearing patterns such as leaf venation, trabecular bone, and spider webs achieve high stiffness per unit mass, yet classical topology optimizers rarely reach such geometries, and few let engineers express structural design intent through natural language. This work treats a frozen text-to-image diffusion model as a training-free source of design knowledge and distills it into the physics loop of density-based topology optimization via score distillation sampling, so that a text prompt becomes an explicit, machine-interpretable representation of engineer intent. The prompt-induced generative gradient and the finite element sensitivity are combined at every iteration, letting physics decide which prompt-induced features survive. In 245 primary SDS runs spanning four geometric domains and two physics regimes, 38 of 49 prompt--domain combinations achieved statistically significant compliance reductions (up to $-31.5\%$ mechanical and $-23.0\%$ thermoelastic), outperforming gradient-based baselines. Cross-domain morphological analysis identifies a recurring structural signature of improvement: in most domains the generative prior suppresses dead-end branches in the rib skeleton, with endpoint--compliance correlation $r = +0.56$ to $+0.99$. A Heaviside projection with $\beta$-continuation resolves a pronounced intermediate-density tendency in this diffusion--physics coupling ($42.6\%$ to $<3\%$), and an automated skeleton-based pipeline converts optimized density fields into \rev{candidate geometry ready for computer-aided design. By retargeting the generative prior across domains, loading conditions, and physics objectives through a change of text prompt, with each new problem's physics setup specified separately, the framework uses a pretrained generative model as a reusable, training-free prior for engineering design.

Qualitative results and an illustration of our core idea. Top left: reconstruction results on benchmark images. Top right: reconstruction results on real-world images. Bottom: illustration of reconstruction-guided noise initialization and modulation. Given multiple input images, we predict a point cloud in canonical space, deterministically inject the predicted geometry into the diffusion process through noise inversion, and modulate the resulting noise to preserve the generative flexibility required to complete unobserved regions and refine visible geometry.

Stochastic transport describes physical systems in which an initially structured distribution spreads under unresolved forcing, scattering, or heterogeneous media. Useful surrogates for such systems should be probabilistic, time-resolved, and able to represent non-Gaussian distributional structure. Generative diffusion models, which corrupt data with Gaussian noise and learn a reverse flow back to structured states, have these properties. Their noise schedules, however, are usually chosen heuristically: image and audio generation---the canonical use cases---provide no physical clock. In transport, by contrast, the variance, or mean-square displacement, is often known from macroscopic theory or empirical scaling even when the full distribution is not. Here we prescribe the forward noising rate as the time derivative of this variance, turning generative time into a calibrated transport clock. The variance path is enforced by construction, while the learned score field represents how non-Gaussian structure inherited from entrance data is smoothed along that path, requiring no intermediate-time physical transport data. For ballistic-to-diffusive transport in turbulent plasmas, the surrogate matches test-particle distributions, reproduces the laboratory-measured variance scale, and tracks the simulated kurtosis evolution without schedule tuning, enabling calibrated emulation and likelihood-based inference.

Uniform-state discrete diffusion models update all tokens in parallel while keeping every position revisable. Even when the commonly used top-$p$ rule leaves only one candidate at a position, that choice affects only the current reverse step and can be revised at the next sampling step. We ask what changes when selected hypotheses instead become persistent context for later predictions. We therefore propose committed reveal sampling (CRS), a training-free sampler that stores selected argmax tokens and inserts them into subsequent model inputs. Our analysis gives a rationale for selecting later and for keeping selected tokens visible. Under the exact forward process, the Bayes error of selecting a clean token cannot increase as noise decreases, while in a simple latent-mode model, keeping the selected token visible helps later parallel predictions agree on the same sequence-level choice. Empirically, paired experiments on Duo-distilled then separate this persistent effect from single-step top-$p$ restriction and scalar temperature scaling. Under the same finalization rule, CRS without top-$p$ truncation reaches lower generative perplexity (GenPPL) than fixed $p=0.95$ and $p=0.9$ baselines across budgets of 8--64 function evaluations (NFE). At 64 NFE, the comparison at matched unigram entropy also gives lower GenPPL for CRS, yielding a more favorable GenPPL--entropy tradeoff. Base Duo shows the same direction in a descriptive comparison, while other diversity and continuation metrics can rank these operating points differently. These results identify support restriction and persistent context as distinct controls of that tradeoff.

We study generative modeling of spatial point processes (SPP), where both the number of points and their spatial configuration are governed by a joint distribution. While diffusion models have achieved strong performance in modeling complex distributions, extending them to variable-cardinality SPP remains challenging. Existing approaches either decouple the modeling of cardinality and spatial structure, or rely on discrete trans-dimensional operations to modify the number of points, resulting in inflexible and asymmetric generative dynamics. We propose the existence-field diffusion model (EFDM) for spatial point processes modeling, where each potential point is associated with an existence variable representing its degree of presence. This enables a unified diffusion process that jointly models both spatial locations and cardinality without requiring explicit discrete transitions. We demonstrate that our approach provides a flexible and general framework for generative modeling of spatial point processes, achieving improved modeling capability on datasets with varying cardinality.

The vast chemical design space and complex, interdependent design variables make catalyst discovery for targeted properties highly labor- and resource-intensive. Although generative models have emerged as a promising solution, existing approaches are generally limited to single-property conditioning or narrow chemical spaces. Here, we present Catalyst Diffusion Transformer (CatDiT), a unified framework for inverse catalyst design that generates valid and novel structures ranging from intermetallic alloys to oxide surfaces. By learning compressed latent representations, CatDiT enables efficient training and rapid sampling while supporting simultaneous conditioning on adsorbate type, binding energy, and catalyst class. The model provides reliable control of discrete properties and directional control of continuous properties, enriching candidate pools for reaction-specific catalyst discovery. As a representative application, multi-conditional generation for the nitrogen reduction reaction (NRR) yields 28 density functional theory (DFT)-relaxed alloy candidates that satisfy the target activity window and lie above the pure-metal *N-*H scaling line, corresponding to a ~1.5-fold enrichment over the source distribution. These results establish CatDiT as a practical and scalable approach for property-directed catalyst inverse design and targeted catalyst generation.

Mobile-health interventions increasingly use online learning and decision making algorithms to personalize when to nudge users toward healthier behavior, but a poorly designed algorithm can burden and disengage participants. New algorithm design decisions should therefore be vetted against realistic simulated users before each real-life deployment. We propose a method to develop ``JITAI-Twins'': digital twins of a target subpopulation for comparing candidate online algorithms before a just-in-time adaptive intervention (JITAI) deployment. The method builds on a conditional time-series diffusion model that is temporally consistent (future actions do not affect the generated past), and it supports repeated updating from three sources of information, in three steps: pre-training on a large observational dataset, fine-tuning on small prior intervention deployments in related populations, and inference-time calibration to the next target population from domain-scientist expertise. We validate the twin at each pre-deployment stage of the long-running HeartSteps series (v2 through v4) of physical-activity suggestion intervention deployments, treating each successive deployment as an upcoming study. The proposed method reproduces the target subpopulation's temporal and between-participant structure better than simpler simulators. These results suggest that our twin can be used to simulate a target deployment before it runs, the prerequisite for testing and informing online algorithm design decisions.

Molecular optimization often starts from a pretrained generative model that captures a broad prior over valid molecular structures. At test time, however, the goal is not to sample from this prior, but to use a limited oracle budget to shift generation toward task-specific high-reward molecules. We study this adaptation problem for discrete diffusion models. Each online round couples several choices. The loop must decide which candidates to evaluate, how rewards become model updates, which feedback to reuse, and how far to move beyond the pretrained prior. These choices have mostly been studied in isolation, leaving open whether they complement one another, become redundant, or interfere inside a full online adaptation loop. We conduct controlled studies across six small-molecule binding-affinity tasks and three protein-fitness tasks. We find that acquisition, reward shaping, and model debiasing provide complementary routes to higher reward, especially for small molecules. Replay further stabilizes learning, while validity penalties keep small-molecule exploration on the valid molecular manifold. Together, these findings point to a practical recipe for feedback-efficient molecular optimization: online fine-tuning with acquisition, reward shaping, debiasing, replay, and validity control. This recipe outperforms offline fine-tuning and inference-time search baselines under matched oracle-call budgets and GPU-hour accounting. The gains are largest when high-reward candidates require larger shifts from the pretrained prior.

A single panorama captures the full visual sphere from one camera center, yet confines users to looking around in place without enabling true scene exploration. Converting a single panorama into a persistent, renderable 3D representation for free-viewpoint navigation has attracted growing interest; existing methods either adopt iterative per-view completion that propagates inpainting results to update the underlying geometry, leading to progressive error accumulation and cumbersome multi-step pipelines, or leverage the temporal consistency priors of video generation models, yet the continuous-trajectory constraint intrinsic to such models limits their flexibility in covering scenes from multiple directions simultaneously. We present Pano2World, which takes a single indoor panorama as input and directly outputs a persistent, explorable 3D Gaussian scene. Given the source panorama, Pano2World first reconstructs a coarse 3D Gaussian proxy and renders it at adaptively sampled nearby poses to obtain geometrically aligned guidance panoramas; a panoramic diffusion model then jointly denoises all target views via View-Aware Attention Routing, where each target view simultaneously receives geometric constraints from its corresponding guidance panorama and global semantic guidance from the source panorama, naturally enforcing cross-view consistency. To avoid the information loss incurred by decoding the multi-view hidden features formed during joint denoising back to the pixel domain via VAE, we introduce Latent Feature Adapter, a geometry-aware bridge module that directly distills these hidden features into a scene latent, subsequently decoded into the final 3D Gaussian scene. Experiments demonstrate that Pano2World significantly outperforms existing methods on the multi-position panoramic novel-view synthesis benchmark.

In this paper, we are the first to examine the correlations between class frequency and the multi-scale noise schedule within diffusion models. For score-based generative models, low-density regions often lead to inaccurately estimated scores, thereby compromising the generation quality. Although the multi-scale noise schedule can alleviate this issue during the diffusion process, low-frequency classes still face the challenge of large low-density regions, resulting in more inaccurate estimated scores than high-frequency classes. Furthermore, high-frequency classes tend to dominate the score space, causing a convergence of most data points towards generating samples from these classes. Consequently, samples generated within low-frequency classes exhibit suboptimal quality and limited diversity. To address this challenge, we propose the \textit{Class-frequency Guided (CFRG)} noise schedule, leveraging the insight that low-frequency classes should be endowed with larger-scale noises. To illustrate the effectiveness of our method, we conduct experiments on various tasks, including image generation, image classification, and text-to-image generation, using imbalanced datasets, \textit{i.e.}, CIFAR-100-LT, and ImageNet-LT. By employing the CFRG noise schedule, we achieve substantial improvements over baselines, manifesting the crucial role of frequency statistics in noise schedule design.

Is the uniform-state diffusion framework a more powerful paradigm for discrete diffusion? Recent studies indicate that this may be the case. In combination with predictor-corrector samplers, uniform-state diffusion models (USDMs) produce samples of higher-quality than masked diffusion models (MDMs), and USDMs equal or outperform MDMs in downstream tasks, even though they exhibit greater perplexity. Two issues remain unresolved. First, existing work compares uniform and masked diffusion with un-informed correctors that re-inject noise at random positions, rather than targeting tokens most likely to be wrong. Second, prior work compares full-sequence diffusion models, so we do not know whether the same conclusion holds when tokens are generated block by block. To address these issues, we introduce BlockGen, a blockwise sequence model that we instantiate with both masked and uniform diffusion. BlockGen trains on a mixture of block sizes and its likelihood interpolates between AR and pure diffusion more finely than models with a fixed block size. BlockGen enables AR-informed predictor-corrector sampling (ARPC), which combines AR and diffusion predictions to re-generate unlikely tokens without an auxiliary verifier. Under ancestral sampling, uniform outperforms masked in the block-by-block setting, especially in the few-step regime. Under ARPC, the gap closes and reverses at high NFE. With block size $16$ on GSM8K, MDMs reach slightly higher accuracy than USDMs, and we observe a similar trend in Generative Perplexity on OpenWebText. Find our code at https://github.com/jdeschena/blockgen.

Discrete diffusion models have achieved strong empirical performance in text and other symbolic domains, but, especially for uniform-rate models, they often require many steps to generate a single sample. Existing acceleration methods either rely on training additional quantities or suffer from slow mixing. In this work, we propose a novel Gibbs-based corrector for discrete diffusion models, termed Gibbs-Accelerated Discrete Diffusion (GADD). GADD leverages the structure of the concrete score function to construct Gibbs posterior likelihoods directly, without requiring any additional training beyond standard score estimation. We show that GADD achieves an overall sampling complexity of $\mathcal{O}(\mathrm{polylog} (\varepsilon^{-1}))$, yielding the first such rate for diffusion-based samplers for uniform-rate discrete diffusion models. We also conduct numerical experiments demonstrating the practical advantages of GADD across synthetic data, zero-shot text sampling, and zero-shot conditional music generation. These results corroborate the theory and show that GADD consistently improves sample quality and wall-clock efficiency over standard baselines, including vanilla Euler methods and CTMC correctors. Beyond this, our theoretical analysis introduces a novel framework for analyzing predictor-corrector methods in discrete diffusion models, which may be of independent interest. Unlike existing approaches that rely on the Girsanov change-of-measure technique, our method is based on an induction argument that tracks error propagation across predictor iterations while accounting for inaccuracies in the corrector updates.

Diffusion models have emerged as a dominant framework for generative modeling, but their mathematical foundations are often presented separately through diffusion probabilistic models, score-based modeling, stochastic differential equations, and numerical sampling methods. We write this tutorial to provide a unified and self-contained account of these viewpoints from the perspective of differential equations. Starting from a conditional Gaussian noising process, we derive ordinary differential equation (ODE) and stochastic differential equation (SDE) representations, pass to the corresponding marginal forward dynamics, and then obtain the reverse-time SDE and probability-flow ODE that make generation possible. We show that the central unknown quantity in reverse sampling is the marginal score, explain how score matching becomes the standard denoising objective under a noise-prediction parameterization, and discuss practical reverse-time sampling and guidance. We further place DDPM, DDIM, flow matching, and score-based SDEs in a common framework, and conclude with diffusion language models in continuous embedding space together with a brief discussion of discrete masked-token diffusion. The tutorial is intended as a bridge between the analytical foundations of diffusion processes and the modern generative algorithms built upon them.

Proteins perform their biological functions through three-dimensional structures encoded by amino acid sequences, and ligand-binding protein co-design requires models that generate sequence-structure compatible proteins under explicit ligand constraints. Although continuous diffusion and flow-based models support ligand-aware design in coordinate or latent spaces, existing discrete diffusion protein language models mainly operate over sequence or structure tokens without direct small-molecule conditioning. We introduce \textbf{ProtLiD$^2$}, a \textbf{Prot}ein \textbf{L}igand-conditioned \textbf{D}iscrete \textbf{D}iffusion model for protein sequence-structure co-design. ProtLiD$^2$ jointly generates amino-acid sequence and discrete structure tokens while incorporating ligand chemical and geometric information through geometry-aware cross-attention. Trained on over one million ligand-protein complexes, ProtLiD$^2$ extends masked discrete diffusion to ligand-aware functional protein design. We further propose maximum confidence-margin guided ReMask decoding, an inference-time self-correction strategy that retains confident predictions and remasks uncertain tokens. ProtLiD$^2$ improves global fold confidence over Complexa in whole-protein design, increasing TM-score from 0.672 to 0.802 and pLDDT from 64.55 to 73.00. In pocket co-design, ProtLiD$^2$ reduces active-site BB-RMSD from 3.46/3.40Å for FAIR/PocketGen to 1.97Å, and improves ligand-aware pass rates over PocketGen from 14.86% to 59.73% and from 6.08% to 23.49% under stricter docking thresholds. These results support ligand-conditioned discrete diffusion as an effective token-space framework for functional protein co-design. Code will be available at https://github.com/auroua/ProtLiD.

Protein function is often controlled by ligands that bias the direction of state transitions, such as agonists and antagonists, rather than stabilizing a single conformation. This is especially important for clinically relevant G protein-coupled receptors (GPCRs), where therapeutic efficacy depends on functional directionality. Structure-based design methods optimize binding to static conformations and cannot represent non-reversible, directional effects or systematically distinguish agonist from antagonist behavior. To address this gap, we introduce Transition-Directed Discrete Diffusion for Allosteric Binder Design (TD3B), a sequence-based generative framework that designs binders with specified agonist or antagonist behavior via a directional transition control objective. TD3B combines a target-aware Direction Oracle, a soft binding-affinity gate, and amortized fine-tuning of a pre-trained discrete diffusion model, enabling targeted agonist and antagonist generation decoupled from binding affinity and unattainable by equilibrium-based or inference-only guidance baselines. The code and checkpoints are available at https://huggingface.co/ChatterjeeLab/TD3B.

Discrete diffusion models generate structured sequences by progressively unmasking tokens, but enforcing global property constraints during generation remains an open challenge. We propose primal-dual guided decoding, an inference-time method that formulates constrained generation as a KL-regularised optimisation problem and solves it online via adaptive Lagrangian multipliers. At each denoising step, the method modifies token logits through an additive, constraint-dependent bias, with multipliers updated by mirror descent based on constraint violation. The bias arises as the optimal KL-regularised projection of the constraint, so the constrained distribution remains as close as possible to the model's unconstrained distribution while still satisfying the constraint. The method requires no retraining and no additional model evaluations beyond standard sampling, supports multiple simultaneous constraints, and provides formal bounds on constraint violation. We evaluate our approach on topical text generation, molecular design, and music playlist generation, showing that a single algorithm instantiated via domain-specific scoring functions improves constraint satisfaction while preserving relevant domain-specific quality metrics.

Recent deep learning approaches seek to automate CAD creation by representing a model as a sequence of discrete commands and parameters, and then generating them using autoregressive models or continuous diffusion operating in Euclidean embedding space. However, continuous diffusion perturbs representations in a continuous Euclidean domain that does not reflect the inherently discrete and heterogeneous nature of CAD tokens, often producing perturbed representations that map to semantically invalid symbols. To overcome this limitation, we propose a cascaded discrete diffusion framework for CAD generation, which consists of a command diffusion for generating CAD commands and a parameter diffusion conditioned on CAD commands. Unlike isotropic Gaussian perturbation, the forward process of our approach operates directly over categorical token distributions using delicate transition matrices. For commands, we adopt an absorbing-state transition matrix that progressively corrupts tokens to a designated symbol; for parameters, we introduce specific transition matrices tailored to heterogeneous attributes: a Gaussian kernel for coordinate continuity, a scale-invariant kernel for dimensional values, and a prior-preserving kernel for boolean attributes. The reverse process is achieved by two denoising networks: a Transformer-based encoder for command recovery, and a parameter network with extra local self-attention for command-level interaction and cross-attention for conditional injection. Experiments on the DeepCAD dataset show that the proposed approach surpasses existing autoregressive and continuous diffusion models on unconditional generation metrics, while qualitative results validate effective controllability in conditional generation tasks. Source codes will be released.

When do language diffusion models memorize their training data, and how to quantitatively assess their true generative regime? We address these questions by showing that Uniform-based Discrete Diffusion Models (UDDMs) fundamentally behave as Associative Memories (AMs) $\textit{with emergent creative capabilities}$. The core idea of an AM is to reliably recover stored data points as $\textit{memories}$ by establishing distinct basins of attraction around them. Historically, models like Hopfield networks use an explicit energy function to guarantee these stable attractors. We broaden this perspective by leveraging the observation that energy is not strictly necessary, as basins of attraction can also be formed via conditional likelihood maximization. By evaluating token recovery of $\textit{training}$ and $\textit{test}$ examples, we identify in UDDMs a sharp memorization-to-generalization transition governed by the size of the training dataset: as it increases, basins around training examples shrink and basins around unseen test examples expand, until both later converge to the same level. Crucially, we can detect this transition using only the conditional entropy of predicted token sequences: memorization is characterized by vanishing conditional entropy, while in the generalization regime the conditional entropy of most tokens remains finite. Thus, conditional entropy offers a practical probe for the memorization-to-generalization transition in deployed models.

Discrete diffusion models based on continuous-time Markov chains (CTMCs) have shown strong performance on language and discrete data generation, yet existing approaches typically parameterize the reverse rate matrix monolithically -- through proxies such as concrete scores (SEDD) or clean-data predictions (MDLM, GIDD) -- rather than aligning the parameterization with the intrinsic CTMC decomposition into jump timing and jump direction. We propose \textbf{Neural CTMC}, which exploits the underlying Poisson structure of CTMC dynamics by separately parameterizing the reverse process through an \emph{exit rate} (when to jump) and a \emph{jump distribution} (where to jump) via two dedicated network heads. We show that the evidence lower bound (ELBO) reduces to a path-space KL divergence between the true and learned reverse processes that factorizes into a Poisson KL for timing and a categorical KL for direction, and admits a tractable, gradient-equivalent and consistent loss. Experimentally, scored by Gemma2-9B, our pure-uniform Neural CTMC achieves $16.36$ generative perplexity on TinyStories (vs.\ GIDD $37.60$ and MDLM $42.66$). On OpenWebText, it attains the best perplexity at the same training-token budget across 16--128 sampling steps among the methods we compare (e.g., at 128 steps: Neural CTMC $183.6$ vs.\ MDLM $210.5$ and GIDD $249.8$). To facilitate reproducibility, we release our pretrained weights at https://huggingface.co/Jiangxy1117/Neural-CTMC.

We study reinforcement learning for controlled diffusion processes with unbounded continuous state spaces, bounded continuous actions, and polynomially growing rewards: settings that arise naturally in finance, economics, and operations research. To overcome the challenges of continuous and high-dimensional domains, we introduce a model-based algorithm that adaptively partitions the joint state-action space. The algorithm maintains estimators of drift, volatility, and rewards within each partition, refining the discretization whenever estimation bias exceeds statistical confidence. This adaptive scheme balances exploration and approximation, enabling efficient learning in unbounded domains. Our analysis establishes regret bounds that depend on the problem horizon, state dimension, reward growth order, and a newly defined notion of zooming dimension tailored to unbounded diffusion processes. The bounds recover existing results for bounded settings as a special case, while extending theoretical guarantees to a broader class of diffusion-type problems. Finally, we validate the effectiveness of our approach through numerical experiments, including applications to high-dimensional problems such as multi-asset mean-variance portfolio selection.

Diffusion models generate conditional samples by progressively denoising Gaussian noise, yet the denoising trajectory can stall at visually plausible but low-quality outcomes with conditional misalignment or structural artifacts. We interpret this behavior as local optima in a surrogate quality landscape: Once early denoising commits to a suboptimal global structure, later steps mainly sharpen details and seldom correct the underlying mistake. While existing inference-time approaches explore alternative diffusion states via re-noising with fixed strength or direction, they exhibit limited capacity to escape steep quality plateaus. We propose Controlled Random Zigzag Sampling (Ctrl-Z Sampling),a scalable sampling strategy that detects plateaus in quality landscape via a surrogate score, and allocates exploration only when a plateau is detected. Upon detection, Ctrl-Z Sampling rolls back to noisier states, samples a set of alternative continuations, and updates the trajectory when a candidate improves the score, otherwise escalating the exploration depth to escape the current plateau. The proposed method is model-agnostic and broadly compatible with existing diffusion frameworks. Experiments show that Ctrl-Z Sampling consistently improves generation quality over other inference-time scaling samplers across different NFE budgets, offering a scalable compute-quality trade-off. Code available at: https://github.com/ShunqiM/Ctrl-Z-Sampling.

Diffusion models have proven to be a flexible and effective framework for modelling probability distributions on finite-dimensional spaces. However, many physical modelling problems such as time series are naturally described over function spaces. In this work we apply diffusion models to such stochastic processes. To do so we consider a spectral representation of the data, obtained using a kernel, thereby dissociating the stochastic part of the processes from their space-time structure. As a result, the stochasticity of the processes is entirely encoded in the spectral coefficients, which we truncate and model using standard finite-dimensional diffusion models. By truncating the representation in the spectral domain we ensure our resulting model defines valid stochastic processes, thereby naturally satisfying consistency and exchangeability criteria. Projecting our spectral diffusion models back to the original input space, we show that for any given marginals our approach corresponds to a diffusion model with correlated noise, with explicit covariance matrix given by the kernel. We demonstrate our method's effectiveness for modelling various multimodal datasets as well as conditional sampling by amortising our models with respect to a context set.