Score-Based Generative Modeling through Anisotropic Stochastic Partial Differential Equations

Recent years have seen rapid progress in diffusion-based generative models, notably DDPM and SBGM, which model complex data distributions via progressive noise addition. However, these models primarily use isotropic diffusion, which tends to erase geometric structures like edges and contours, leading to blurred or distorted outputs. Efforts to incorporate structure-awareness include static anisotropic diffusion and learned priors, but these approaches lack dynamic adaptation during the generative process. Classical PDEs, such as Perona–Malik, excel at edge-preserving smoothing but are limited in deep generative contexts. Combining deep learning with nonlinear, adaptive PDEs offers a promising pathway to improve structural fidelity in generated images.

Existing diffusion models often struggle to preserve geometric features like edges during the forward noising process, resulting in artifacts and loss of detail. The challenge lies in designing a structure-aware forward process that can dynamically adapt to local image features, maintaining edges over longer timescales while still enabling effective reverse sampling. Achieving this balance is difficult because static anisotropic diffusion cannot respond to evolving structures, and linear models lack the flexibility to model complex, nonlinear geometric transformations. Addressing this problem is crucial for high-fidelity image synthesis, especially in applications demanding detailed structural accuracy.

The core innovation is the formulation of a nonlinear, time-dependent anisotropic SPDE that combines structure-sensitive drift and diffusion terms. This allows the forward process to adaptively preserve edges and contours by modulating anisotropy coefficients based on current image structure. Unlike prior static anisotropic models, this approach dynamically adjusts anisotropy, enabling longer retention of geometric cues. The integration of neural score estimation with discretized SPDEs provides a robust framework for high-quality reverse sampling. This method bridges classical PDE techniques with modern deep learning, offering a flexible, theoretically grounded solution for structure-aware image generation.

• �� Define the forward process as a nonlinear SPDE with drift \(b(t, u) =

abla \cdot g_1(t,

abla u)

abla u\) and diffusion \(\sigma(t, u) v = g_2(t,

abla u) v\).• Design spatially varying coefficients \(\alpha_i(t)\) and \(\lambda_i(t)\) controlling destruction strength and anisotropy, with \(\lambda_1(t)\) and \(\lambda_2(t)\) modulating edge preservation and noise directionality.• Discretize the SPDE using finite difference or Galerkin methods, transforming it into a finite-dimensional SDE for simulation.• Train neural networks to approximate the score function \(

abla \ln p_t\) of the perturbed data distribution, leveraging the structure-aware process.• Implement reverse sampling by solving the corresponding time-reversed SDE, guided by the learned score, to generate high-quality images with preserved geometric features.

Experiments utilize CIFAR-10 and CelebA-HQ datasets, evaluating image quality via FID and Inception Score. The models compare isotropic versus anisotropic formulations, with hyperparameters tuned for optimal structure preservation. Ablation studies analyze effects of \(\alpha_i(t)\) and \(\lambda_i(t)\) variations. Training employs Adam optimizer with learning rate 1e-4 over 200 epochs. Sampling efficiency and quality are assessed through qualitative visualizations and quantitative metrics, with particular focus on edge clarity and detail retention. Conditional stroke-to-image tasks further demonstrate the model’s ability to maintain structural integrity under guided generation.

The anisotropic SPDE model achieves 8% lower FID scores on CIFAR-10 and 5% on CelebA-HQ compared to isotropic baselines, with faster convergence. Visual results show sharper edges, clearer contours, and finer details. In stroke-to-image tasks, the model reduces reconstruction error by approximately 10%, producing more faithful geometric reproductions. Ablation results confirm that dynamic anisotropy coefficients significantly contribute to structure retention, validating the theoretical advantages of the proposed approach.

This framework is suitable for high-fidelity image synthesis, medical imaging reconstruction, and artistic content creation, especially where geometric accuracy is vital. It can be integrated into existing pipelines for style transfer, super-resolution, and virtual environment generation. The structure-preserving property enhances applications in domains requiring detailed and realistic visuals, such as radiology, digital art, and immersive simulations.

Parameter tuning complexity remains a challenge, requiring careful calibration of anisotropy coefficients. Numerical discretization introduces approximation errors, potentially affecting theoretical guarantees. High computational costs, especially at high resolutions, limit real-time applications. Future work should focus on adaptive parameter learning, efficient numerical schemes, and extending the framework to 3D and video data.