Title: Discrete Spatial Diffusion: Intensity-Preserving Diffusion Modeling
Venue: NeurIPS 2025
Accepted: yes
Keywords: Diffusion models

physics

spatial diffusion

continuous-time discrete-state Markov process

mass conservation

intensity conservation

scientific data
Abstract: Generative diffusion models have achieved remarkable success in producing high-quality images. However, these models typically operate in continuous intensity spaces, diffusing independently across pixels and color channels. As a result, they are fundamentally ill-suited for applications involving inherently discrete quantities such as particle counts or material units, that are constrained by strict conservation laws like mass conservation, limiting their applicability in scientific workflows. To address this limitation, we propose Discrete Spatial Diffusion (DSD), a framework based on a continuous-time, discrete-state jump stochastic process that operates directly in discrete spatial domains while strictly preserving particle counts in both forward and reverse diffusion processes. By using spatial diffusion to achieve particle conservation, we introduce stochasticity naturally through a discrete formulation. We demonstrate the expressive flexibility of DSD by performing image synthesis, class conditioning, and image inpainting across standard image benchmarks, while exactly conditioning total image intensity. We validate DSD on two challenging scientific applications: porous rock microstructures and lithium-ion battery electrodes, demonstrating its ability to generate structurally realistic samples under strict mass conservation constraints, with quantitative evaluation using state-of-the-art metrics for transport and electrochemical performance.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe article is about generating artificial images (or other data) by reversing a random process with a neural network. Image generation with a Gaussian diffusion reversing neural network is a popular topic. In this process, an image is generated from Gaussian noise by iteratively reversing the diffusion. In this process, the pixel values of the images are supposed to be continuous. \n\nThe article proposes a new random process in which the image (or data) has discrete values, with the constraint that the total mass (sum of all pixel values) is preserved between each random step in each color channel. This process can be applied to RGB images because they are originally uint8, but it is particularly suited for generating other types of data, such as mineral tomography. This data is binary, and controlling the amount of porosity (number of zeros) and the shape of the cavities is important so that they look natural. Thus, porosity can be controlled at initialization by the number of zeros in the initial random image.\n\nThe article presents the forward discrete mass-preserving process in detail. This process can be represented in two ways: with the random positions of individual particles or with population histograms of each cell in the image. In the forward process, each particle has an equal probability of jumping into one of its four neighboring cells or of not moving. The backward process is similar but different. Because it is conditioned by the initial position of the particle, the directions are no longer equiprobable, as the particle will be pushed to return to its initial state.\n\nAs with Gaussian diffusion, the discrete process is reversed using a network. The network is trained to estimate the probability that a particle will reverse jump in each direction. The authors propose two loss functions: one that minimizes the L1 distance between the network guess and the true backward process and one that maximizes the likelihood of the network in the backward process.\n\nIn the \"Experiment\" section, the authors demonstrate that their process can generate RGB images similar to those produced by Gaussian diffusion. Then, they present results on tomography generation, demonstrating that their method enables them to control porosity while producing realistic data.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths :\n\nThe article proposes an interesting alternative to Gaussian diffusion. The random forward and backward processes are explained quite well. The authors demonstrate that this process works on natural RGB images, even though that is not their main target application. They also acknowledge the limitations, explaining that this process is not efficient for RGB images because the number of random variables is higher than in Gaussian diffusion.\n\nThe article is well-written and covers all aspects of this type of solution (random process, forward-backward, loss, sheduler ..), with extensive citation of applicative and theoretical articles.\n\nThe article also provides extensive supplementary material, including all the necessary mathematical derivations and additional experiments. \n\nIn this sense, I find that the article provides a thorough description of the proposed method.\n\nWeaknesses :\n\nEven though the article proposes many interesting experiments, I think an important one is missing. The article claims that Gaussian diffusion is not suited for tomography generation and that the proposed process is better. Therefore, I think the experiment should highlight this fact by comparing the proposed solution with tomographies generated using the Gaussian process  ( with a conditioning on porosity and relaxation from discrete/binary to continuous). \n\nIn my opinion, since the article describes a random process, it should use more common probabilistic notations and definitions. This would make it easier to compare to other methods of this kind.\n\nQUESTIONS:\n(0) My main concern is that the results of classical Gaussian diffusion should be provided for comparison in experiments. Even if the results are similar, this new framework will still be interesting.\n\n(1) Regarding my criticism of the notation, I think it would be clearer if the forward equations (1a, b) and backward equation (4) used probabilistic notation and definitions. For example, I think it would be better to describe the  \"noise\" $\\nu$ as categorical random variables with:\n\\begin{equation}\n\\mathbb{P}(\\nu) = \\begin{cases}\nr & \\text{if } \\nu\\in \\{(0,1),(0,-1),(1,0),(-1,0)\\};\\\\\n1-4r & \\text{if } \\nu = (0,0); \\\\\n0     & \\text{otherwise}\n\\end{cases}\n\\end{equation}\n\nIt may help to understand the likelihood in equation (6). Personally, I have trouble seeing where it comes from.\n\n(2) Table 3 should be cited in the text of the experiments so that readers know the experimental setup is provided in the supplementary material.\n\n(3) Equations (5), (6), and (7) should state the network inputs and variables on which the sums and integrals are based to make them easier to understand. For example, in equations (6) and (7), it is unclear what the sums are.\n\n(4) line 657, I think there is a typo and that $p_{i,l}$ should be replaced with $p_{k,l}$.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a discrete spatial diffusion (DSD) model, in which multiple particles are distributed in space and their total number is preserved throughout both the forward and reverse processes. This conservation property allows the model to control the number of generated particles by specifying the initial count in the reverse process. In both diffusion directions, each particle stochastically transitions to a neighboring region. A deep neural network (DNN) is employed to learn the transition probabilities at each time step of the reverse process. The training objective is conditioned on the observed data $I_0$ and is formulated analogously to standard diffusion models. The authors validate the effectiveness of the proposed approach through experiments on common image datasets (MNIST, CIFAR10, CelebA) and domain-specific data such as rock microstructures and lithium-ion battery electrodes.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n* A novel approach to modeling spatial distributions of particles using diffusion models while preserving the total number of particles.\n* Experimental validation on real-world applications, including rock microstructures and lithium-ion battery electrodes.\n\n### Weaknesses\n* Certain aspects of the forward and reverse processes remain unclear (see Questions below).\n* As noted by the authors, computational cost becomes a critical issue in high-intensity scenarios.\n\nQUESTIONS:\n### Questions\n\n* Regarding Eq. (1a):\n    * This equation appears to describe the transition of a single particle in the forward process, with $r$ referred to as the transition rate. However, in Table 8, values such as $r = 120$ are used. What does it mean for the transition rate to be $r = 120$?\n    * If $r$ represents the number of particles transitioning from $I_{x,y,c}$, how is the transition handled when $I_{x,y,c} < r$?\n    * Does the model assume that transitions are made with equal probability to the four neighboring directions included in $\\nu$ (or $\\bar{\\nu}$ in Eq. (3))?\n\n\n### Comments\n* In Appendix H, the authors state that “we found that our model was very robust with respect to the hyperparameters used.” It would be helpful to include a comparison of the forward and reverse processes under different values of $r$ (e.g., on MNIST), to illustrate this robustness more concretely.\n\n* Throughout the paper, the abbreviation for the Inverse Heat Dissipation Model should be used consistently. It is currently inconsistent between “IHDM” and “IHD,” and should be unified.\n\nLIMITATIONS:\nSee weaknesses above.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel variant of diffusion models that preserves particle counts throughout both the forward and reverse processes, which is a property desired for scientific applications where discrete quantity conservation is essential. The key idea is to reformulate the standard Gaussian corruption process as a continuous-time, discrete-state jump stochastic process that operates directly in the discrete spatial domain. The model’s effectiveness is demonstrated through experiments on two scientific applications, with additional validation conducted on standard image benchmarks for computer vision tasks.\n\nSTRENGTHS AND WEAKNESSES:\n**S1:** The problem addressed is niche but compelling in the context of scientific applications. The paper is well-structured and clearly presented, with both the motivation and core technical contributions effectively communicated.\n\n**S2:** The Markovian jumping process is an interesting and intuitive idea to achieve the preservation of intensity from the formulation level, thus presenting a rather elegant way to solve the challenge.\n\n**S3:** The experiments effectively demonstrate the benefits of the proposed intensity-preserving property, even though this characteristic may not appear immediately critical for general vision tasks.\n\n**W1:** As acknowledged in the limitations section, the proposed DSD framework introduces significant computational overhead compared to standard Gaussian diffusion models, both during training and inference. Given that Gaussian diffusion is already computationally intensive, this further limits the practical adoption of DSD in mainstream vision applications.\n\n**W2:** The experimental setup and evaluation protocols, particularly for the scientific use cases, remain somewhat unclear. For instance, is the goal to synthesize images of subsurface rock microstructures and lithium-ion electrodes from randomly initialized images, as outlined in Algorithm 2? If so, the inference process appears to require predefined condition numbers and intensity values to initiate generation. How feasible or practical is this requirement within real-world scientific workflows? Similarly, on the evaluation side, while the proposed DSD demonstrates improved intensity preservation, are the generated scientific samples rigorously valid within their respective domains? A more detailed discussion on the scientific fidelity and downstream utility of the generated data would strengthen the work.\n\nMinors: There are some formatting inconsistencies across subtitles, for example, some section headers begin with bolded first letters while others do not.\n\nQUESTIONS:\nPlease see the weaknesses mentioned above.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors proposes Discrete spatial diffusion (DSD), a continuous time, discrete-state diffusion framework that preserves total image intensity (i.e., particle counts) in both forward and reverse processes by modeling intensity units as spatially correlated random walks. Their proposed method achieves competitive performance compared to continuous diffusion models (in the natural image domain), and demonstrates strong domain specific performance by accurately reconstructing porous rock microstructures and Li-ion battery electrode morphologies under strict intensity conservation.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. The work has a clear problem motivation, addressing the need for generative models that strictly conserve total image intensity, which is needed for scientific and engineering applications where mass/energy preservation is non‐negotiable.\n2. The proposed method guarantees strict mass conservation per color channel, enabling scientifically meaningful generation under hard constraints. They also rigorously justify their work through theory. The derivations of forward and reverse markov jump processes, noise scheduling via SSIM, and principled loss functions (rate matching) are clear.\n\nWeaknesses\nTheir proposed method's forward and reverse sampling scales linearly with total intensity, leading to slower inference compared to vanilla diffusion models and potentially prohibits runtimes for high bit depth or very high resolution images. However, I acknowledge that these weaknesses are not within the scope of this work, and the authors also make an effor to acknowledge these weaknesses through engineering implementations and a section in their appendix (F.3) that directly addresses that their method remains computationally feasible.\n\nOverall, this work is technically solid with high impact in generative modeling under conservation laws, well-substantiated theory, and strong domain specific results. Minor efficiency and usability tradeoffs do not outweigh its contributions.\n\nQUESTIONS:\nThe custom jump-process and adaptive sampler adds engineering complexity into well established diffusion frameworks. Are there plans to release modular code or wrappers that would make DSD easy to plug into existing open source codebases?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-29"}