Integrating graph neural networks (GNNs) with variational inference (VI) provides a promising direction for blending structured prior knowledge with observational empirical data for data-driven industrial process modeling. However, this task requires inference of the normalized adjacency matrix (NAM), where each row is normalized to be non-negative and to sum to one, matching the support of Dirichlet distribution. This requirement presents two main technical challenges: 1) intractable Kullback-Leibler (KL) divergence optimization between Dirichlet distributions, and 2) constrained optimization for standard- gradient-descent-based neural network parameter optimization. To handle issue 1), we first formulate the inference of the NAM as a differential equation simulation problem and derive an easy-to-implement expression to iteratively improve the KL divergence without explicitly computing it. Based on this, to alleviate issue 2), we involve Riemannian optimization to precondition this simulation procedure, which ensures that the inferred NAM conforms to the row-normalization constraint. After that, we collectively designate these approaches for NAM inference as Preconditioned-Simulation-Induced Variational Inference ($\psi$-VI), and provide theoretical guarantees of convergence. On this foundation, we propose a new graph neural network architecture, the Preconditioned-Simulation-Induced-based Variational Graph Neural Network ($\psi$-VGNN) for industrial process modeling. Finally, we validate the efficacy of$\psi$-VGNN through comprehensive experiments on industrial modeling tasks.

The diffusion models (DMs) have demonstrated the remarkable capability of generating images via learning the noised score function of data distribution. Current DM sampling techniques typically rely on first-order Langevin dynamics at each noise level, with efforts concentrated on refining inter-level denoising strategies. While leveraging additional second-order Hessian geometry to enhance the sampling quality of Langevin is a common practice in Markov chain Monte Carlo (MCMC), the naive attempts to utilize Hessian geometry in high-dimensional DMs lead to quadratic-complexity computational costs, rendering them non-scalable. In this work, we introduce a novel Levenberg-Marquardt-Langevin (LML) method that approximates the diffusion Hessian geometry in a training-free manner, drawing inspiration from the celebrated Levenberg-Marquardt optimization algorithm. Our approach introduces two key innovations: (1) A low-rank approximation of the diffusion Hessian, leveraging the DMs' inherent structure and circumventing explicit quadratic-complexity computations; (2) A damping mechanism to stabilize the approximated Hessian. This LML approximated Hessian geometry enables the diffusion sampling to execute more accurate steps and improve the image generation quality. We further conduct a theoretical analysis to substantiate the approximation error bound of low-rank approximation and the convergence property of the damping mechanism. Extensive experiments across multiple pretrained DMs validate that the LML method significantly improves image generation quality, with negligible computational overhead.

Recent Diffusion models (DMs) advancements have explored incorporating the second-order diffusion Fisher information (DF), defined as the negative Hessian of log density, into various downstream tasks and theoretical analysis. However, current practices typically approximate the diffusion Fisher by applying auto-differentiation to the learned score network. This black-box method, though straightforward, lacks any accuracy guarantee and is time-consuming. In this paper, we show that the diffusion Fisher actually resides within a space spanned by the outer products of score and initial data. Based on the outer-product structure, we develop two efficient approximation algorithms to access the trace and matrix-vector multiplication of DF, respectively. These algorithms bypass the auto-differentiation operations with time-efficient vector-product calculations. Furthermore, we establish the approximation error bounds for the proposed algorithms. Experiments in likelihood evaluation and adjoint optimization demonstrate the superior accuracy and reduced computational cost of our proposed algorithms. Additionally, based on the novel outer-product formulation of DF, we design the first numerical verification experiment for the optimal transport property of the general PF-ODE deduced map.

Diffusion models have driven the advancement of vision generation over the past years. However, it is often difficult to apply these large models in downstream tasks, due to massive fine-tuning cost. Recently, Low-Rank Adaptation (LoRA) has been applied for efficient tuning of diffusion models. Unfortunately, the capabilities of LoRA-tuned diffusion models are limited, since the same LoRA is used for different timesteps of the diffusion process. To tackle this problem, we introduce a general and concise TimeStep Master (TSM) paradigm with two key fine-tuning stages. In the fostering stage (1-stage), we apply different LoRAs to fine-tune the diffusion model at different timestep intervals. This results in different TimeStep LoRA experts that can effectively capture different noise levels. In the assembling stage (2-stage), we design a novel asymmetrical mixture of TimeStep LoRA experts, via core-context collaboration of experts at multi-scale intervals. For each timestep, we leverage TimeStep LoRA expert within the smallest interval as the core expert without gating, and use experts within the bigger intervals as the context experts with time-dependent gating. Consequently, our TSM can effectively model the noise level via the expert in the finest interval, and adaptively integrate contexts from the experts of other scales, boosting the versatility of diffusion models. To show the effectiveness of our TSM paradigm, we conduct extensive experiments on three typical and popular LoRA-related tasks of diffusion models, including domain adaptation, post-pretraining, and model distillation. Our TSM achieves the state-of-the-art results on all these tasks, throughout various model structures (UNet, DiT and MM-DiT) and visual data modalities (Image, Video), showing its remarkable generalization capacity.

Particle-based Variational Inference (ParVI) methods approximate the target distribution by iteratively evolving finite weighted particle systems. Recent advances of ParVI methods reveal the benefits of accelerated position update strategies and dynamic weight adjustment approaches. In this paper, we propose the first ParVI framework that possesses both accelerated position update and dynamical weight adjustment simultaneously, named the General Accelerated Dynamic-Weight Particle-based Variational Inference (GAD-PVI) framework. Generally, GAD-PVI simulates the semi-Hamiltonian gradient flow on a novel Information-Fisher-Rao space, which yields an additional decrease on the local functional dissipation. GAD-PVI is compatible with different dissimilarity functionals and associated smoothing approaches under three information metrics. Experiments on both synthetic and real-world data demonstrate the faster convergence and reduced approximation error of GAD-PVI methods over the state-of-the-art.

Diffusion models have demonstrated competitive performance in missing data imputation (MDI) task. However, directly applying diffusion models to MDI produces suboptimal performance due to two primary defects. First, the sample diversity promoted by diffusion models hinders the accurate inference of missing values. Second, data masking reduces observable indices for model training, obstructing imputation performance. To address these challenges, we introduce $\underline{\text{N}}$egative $\underline{\text{E}}$ntropy-regularized $\underline{\text{W}}$asserstein gradient flow for $\underline{\text{Imp}}$utation (NewImp), enhancing diffusion models for MDI from a gradient flow perspective. To handle the first defect, we incorporate a negative entropy regularization term into the cost functional to suppress diversity and improve accuracy. To handle the second defect, we demonstrate that the imputation procedure of NewImp, induced by the conditional distribution-related cost functional, can equivalently be replaced by that induced by the joint distribution, thereby naturally eliminating the need for data masking. Extensive experiments validate the effectiveness of our method. Code is available at [https://github.com/JustusvLiebig/NewImp](https://github.com/JustusvLiebig/NewImp).

The inversion of diffusion model sampling, which aims to find the corresponding initial noise of a sample, plays a critical role in various tasks. Recently, several heuristic exact inversion samplers have been proposed to address the inexact inversion issue in a training-free manner. However, the theoretical properties of these heuristic samplers remain unknown and they often exhibit mediocre sampling quality. In this paper, we introduce a generic formulation, \emph{Bidirectional Explicit Linear Multi-step} (BELM) samplers, of the exact inversion samplers, which includes all previously proposed heuristic exact inversion samplers as special cases. The BELM formulation is derived from the variable-stepsize-variable-formula linear multi-step method via integrating a bidirectional explicit constraint. We highlight this bidirectional explicit constraint is the key of mathematically exact inversion. We systematically investigate the Local Truncation Error (LTE) within the BELM framework and show that the existing heuristic designs of exact inversion samplers yield sub-optimal LTE. Consequently, we propose the Optimal BELM (O-BELM) sampler through the LTE minimization approach. We conduct additional analysis to substantiate the theoretical stability and global convergence property of the proposed optimal sampler. Comprehensive experiments demonstrate our O-BELM sampler establishes the exact inversion property while achieving high-quality sampling. Additional experiments in image editing and image interpolation highlight the extensive potential of applying O-BELM in varying applications.