Discrete Spatial Diffusion: Intensity-Preserving Diffusion Modeling

Thank you for the constructive feedback and for recognizing the novelty and thoroughness of our method. We believe the new experimental results and clarifications further strengthen the case for this work’s significance, particularly in scientific domains where conservation laws are important.

[Q0]

Thank you for encouraging us to include a comparison with a Gaussian diffusion framework. This was a valuable suggestion, and we have carried out the experiment accordingly. We conducted the experiment using our Savonnières Carbonate sample, which displays the widest range of heterogeneous geological attributes in our dataset, including dissolution features, fossils, and grain inclusions. The results confirm key limitations of Gaussian diffusion for our target application. We will document this experiment thoroughly in the final version, although we cannot include sample visualizations here, we will do our best to describe the quantitative and qualitative results:

• The generated samples are noticeably “grainy” and exhibit artifacts.

• Grain boundaries are not perfectly smooth or round (compared to our samples in Figure 4 and 15) but appear as staircase-like structures (like those in Minecraft).

• Salt-and-pepper noise is present both inside and outside the solid regions.

• While the porosity (i.e., the mean pixel intensity over image size) is approximately correct (especially near the mean ~0.6), this is achieved through a bimodal distribution centered near 0 and 1, with values falling outside the [0,1] range (e.g., down to –0.2 and up to 1.2). As a result, binarization does not significantly worsen the porosity error, unlike with datasets like MNIST (Figure 14).

• We trained the model using data whose porosity is between 53% and 72%. We deployed the model to a wider range of target porosity between 45.74% and 81.24%. The model confidently produces incorrect predictions (low standard deviation over 100 generated samples) outside of the training distribution, indicating poor generalization outside the center of the distribution. In contrast, DSD was able to generalize, since intensity is a hard constraint in DSD.

These findings highlight that while the Gaussian diffusion model may approximate average porosity reasonably well near the mean, it fails to preserve structure and accuracy in the mean intensity away from this. More importantly, the lack of spatial coherence and the introduction of noise compromises the generated samples. This is especially important because transport properties in porous media are highly sensitive to such artifacts, small variations or noise can clog pores (percolation threshold) and significantly alter effective properties.

[Q1] We thank the reviewer for the suggestion. However, respectfully, we are using standard notation. Since $r$ is the transition rate (not a probability, but probability per unit time), we could not equate the $\mathbb{P}(\nu)$ to $r$ in the reviewer’s suggested notation (more about that on the answer to Reviewer tBZm's questions). Secondly, in the context of chemical reactions, $\nu$ is often to be interpreted as stoichiometric coefficients (which are not random variables) which prescribe the change of chemical species’ populations (in our context, we can think of discrete location $(x,y,c)$ as populations of three chemical species.) We would argue that the notation in the submitted version is the standard notation in existing technical literature, for example, see “Solving the chemical master equation for monomolecular reaction systems and beyond: a Doi-Peliti path integral view” by Vastola and “Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes” by Lin and Buchler. As such, we would like to retain our current model specification. To improve the clarity of the manuscript we will explain the link between the transition rates and jump probabilities in the infinitesimal time limit.

[Q2] We thank the reviewer for their suggestion. We have modified the sentence “The hyperparameters used can be found in Appendix H.” to “The hyperparameters used can be found in Table 3 in Appendix H.”

[Q3] We thank the reviewer for the suggestions. We have inserted the dependence of $\bar{\nu}, x, y, c$ and explicitly annotated the sum and mean over these variables.

[Q4] Thank you. We have fixed the typo.

Thank you for carefully reading of our work. We appreciate your thoughtful questions about the transition dynamics and your attention to detail regarding notation and clarity. Please find our detailed responses to your comments and questions below.

[Questions]

• We follow the standard model specification for continuous-time stochastic processes, for example, in Van Kampen’s Stochastic Processes in Physics and Chemistry, Gardiner’s Stochastic Methods, and Durrett’s Essentials of stochastic processes.

• By definition, a transition rate is the probability of a potential “reaction” to take place per unit time, in the infinitesimal-time limit. For our forward process, $r=1000$ would mean that within time $\Delta t \downarrow 0$, a specific particle will move to its right (as well as left, up, and down) with a probability $p=1000 \Delta t$. The infinitesimal condition is imposed to eliminate multiple jump events, such as two consecutive jumps, which have negligible probability ($\mathcal{O}(\Delta t^2)$) in this infinitesimal time limit. Writing these operations with finite-time probability equations is far more complicated as it requires accounting for multiple-jump events even when the process is generated by simpler single-jump rates. To improve the clarity of the manuscript we will explain this more carefully.

• It is important to point out that in Eq. (1), $r$ stands for the transition for each of the particles located at $(x,y,c)$. Thus, even if there is only one particle, the transition rate is still $r=1000$. Since the transition rate is probability of a reaction to take place per unit time, we cannot compare its numerical value directly to the particle number: Even if we have only one particle at that location, it is fine to have $r=1000$; it just means that the particle has a higher propensity to make a move. In addition, we remark that when there are $I_{x,y,c}$ particles, the total transition rate would be $r \times I_{x,y,c}$ due to independence between the particles; in this case, it states that within time $\Delta t \downarrow 0$, one of the $I_{x,y,c}$ particles will move to its right (as well as left, up, and down) with a probability $p=1000 I_{x,y,c} \Delta t$.

• In terms of the choice of $r$, we chose it so that at time $t=1$, the spatial diffusion process ensures sufficient mixing to a random configuration (so easy to sample during inference; see for example the SSIM metric in the Appendix). We remark that $r$ and terminal time (1 in our case) are co-design choices. We heuristically designed the noise scheduler using the SSIM principle. We observed that the choice of the noise scheduler does not seem to significantly affect the performance of the model.

• Yes, our model uses transitions that are made with equal probability to the four neighboring directions. This is because the particle’s transition rates to each of the directions are equal. Another way to characterize our forward process is that it is a spatially symmetric random walk. That said, we also carried-out experiments with no-flux boundary conditions (reported on Table 3) where the particles cannot transition periodically across the edge of the images. We did not identify discernible differences between the two approaches, showing the robustness of the spatial diffusion formulation.

[Comments]

• Thank you for pointing out the inconsistency in the use of the acronym. We have reviewed and corrected all occurrences to ensure consistent and accurate usage throughout the manuscript.

Thank you for the thoughtful and encouraging review, we're happy that the motivation and theoretical clarity came through.

[Q1] We appreciate the comment. The codebase which we will release is relatively modular and designed using scipy, numpy, pytorch, and the scoremodels python package so as to best make use of existing tools and isolate the complexities associated with DSD in particular. We have routines for a) generating the Greens functions associated with the noising process, b) applying the noising process using dataloader workers, and c) sampling from a trained DSD network, which are where the additional engineering complexities lie within DSD. A user can easily swap in standard diffusion components, such as different architectures, and time (noise) schedules. Datasets in binarized or RGB formats can be incorporated with minimal effort, and it is possible to modify the integration parameters/strategies used during sampling. We will release the self-contained codebase on GitHub upon acceptance.

Thank you for the detailed and constructive feedback, we're glad the motivation and elegance of the approach came through, and appreciate the opportunity to clarify our scientific evaluation protocols.

[W2] Thank you for pointing this out. We apologize that aspects of the scientific dataset evaluation were unclear. For the human-centric datasets, we initialized inference from a random distribution to demonstrate our model’s capacity to generate human-centric RGB images. However, for the scientific domains (here, rock and battery microstructures), the inference allows for more practical use cases. After training, our framework can interpolate across intensity values not seen during training, enabling the exploration of intermediate or extrapolated physical states. For instance, in Appendix F.3 (MNIST), we show that the model produces coherent, semantically valid outputs even under out-of-distribution intensity conditions, suggesting learned representations that generalize and reflect meaningful physical structure.

The ultimate goal of scientific applications is to generate microstructures conditioned on specific parameters for downstream analysis. In earth sciences, the types of samples we present in this work are immensely valuable, not only because acquiring them can cost millions of dollars, but because they offer a window through which we can observe and measure the complexity of the geology below us. Extracting as much information as possible from these scarce datasets, and then being able to generate new realizations, opens up avenues like performing uncertainty quantification, or interpolate to physical regimes that were not directly observed during sampling.

Practically, in the case of rock microstructures, the generated samples could inform decisions in applications such as aquifer remediation, CO₂ sequestration, or hydrogen storage. In battery science, the aim is to generate novel electrode morphologies that are optimized for energy storage performance, subject to physical and material constraints (largely driven by economics). With these in mind, in real-world workflows, summary descriptors like porosity (for rocks) can often be estimated without expensive 3D x-ray imaging for example, through bulk measurements, indirect methods at the field (e.g., well logs), rapid and inexpensive lab-based porosity tests, or conventional 2D imaging, making it feasible to condition generation on realistic parameters. This opens up avenues for generating physically plausible realizations consistent with known constraints.

We also conducted rigorous domain-specific evaluations. For the rock samples, we computed pore size distributions and two-point correlation functions using PoreSpy[1]. These showed excellent agreement between real and generated samples (Appendix F.2, Fig. 16). For battery electrodes, we evaluated interface length, triple-phase boundary, and relative diffusivity using TauFactor[2] (Appendix G.2, Fig. 19), all of which quantitatively confirm the structural and functional fidelity of our generated images. Additionally, we report a significantly improved FID score (0.9 vs. 18.1 in [3]), although we acknowledge FID’s limitations for binary images. These analyses support the scientific validity and potential utility of DSD samples in downstream modeling/design/ and uncertainty quantification tasks.

[1] Gostick, J. T., Khan, Z. A., Tranter, T. G., Kok, M. D., Agnaou, M., Sadeghi, M., & Jervis, R. (2019). PoreSpy: A python toolkit for quantitative analysis of porous media images. Journal of Open Source Software, 4(37), 1296.

[2] Cooper, S. J., Bertei, A., Shearing, P. R., Kilner, J. A., & Brandon, N. P. (2016). TauFactor: An open-source application for calculating tortuosity factors from tomographic data. SoftwareX, 5, 203-210.

[3] Kang-Hyun Lee and Gun Jin Yun. Microstructure reconstruction using diffusion-based generative models. Mechanics of Advanced Materials and Structures, 31(18):4443–4461, 2024.

[Minors] Thank you, we have identified some inconsistencies in the capitalization used within section titles and have resolved the matter; hopefully this is the inconsistency to which you refer. We will make every effort to hunt down any other formatting problems, and if you have further issues to note then just let us know.