The book "The Principles of Diffusion Models" outlines the foundational concepts behind diffusion models, tracing their evolution from a forward process that corrupts data into noise to a reverse process that reconstructs data. It presents three complementary perspectives—variational, score-based, and flow-based—each describing how a time-dependent velocity field transports a simple prior to the data distribution. The text also covers practical guidance for controllable generation, efficient solvers, and diffusion-inspired flow-map models, providing a mathematically grounded framework for readers with basic deep‑learning knowledge.
arXiv:2506. 20771v2 Announce Type: replace Abstract: We propose a latent score-based generative AI framework for learning stochastic, non-local closure models and constitutive laws in nonlinear dynamical systems of computational mechanics.
The paper argues that the concrete random noise used in diffusion models is not merely a passive perturbation but a learnable input that can be exploited by the model. By analyzing how clean data and realized noise jointly form the noisy input, the authors show that the model can learn regularities in the data or in the noise structure, and that these two routes can interact. Experiments on MNIST and CIFAR‑10 using pseudorandom streams demonstrate that structured‑noise training can reduce prediction loss, but this advantage disappears when test noise is replaced with IID noise, indicating that the learned dependence is tied to the specific noise structure.
arXiv:2512. 20963v3 Announce Type: replace Abstract: Diffusion models excel at generating high-quality, diverse samples, yet they risk memorizing training data when overfit to the training objective.
arXiv:2604. 07366v2 Announce Type: replace Abstract: Partial differential equations (PDEs) govern nearly every physical process in science and engineering, but solving them at scale remains prohibitively expensive.
arXiv:2603. 18907v2 Announce Type: replace Abstract: We propose a new Neural Galerkin Normalizing Flow framework to approximate the transition probability density function of a diffusion process by solving the corresponding Fokker-Planck equation with an atomic initial distribution, parametrically with respect to the location of the initial mass.
arXiv:2608. 07161v1 Announce Type: cross Abstract: Simulating complex fluid flows requires capturing full equilibrium distributions rather than just mean trajectories, yet high-fidelity solvers remain computationally prohibitive.
One of the primary challenges in Bayesian inference on the parameters of a diffusion model from discrete observations is the unavailability of an analytical expression for the transition density function between consecutive observation times, which is needed to derive the likelihood function. Extending previous studies that solve Fokker-Planck (FP) type partial differential equations with Normalizing Flows, we propose a new Normalizing Flow architecture to learn the transition density function of the diffusion process between two observation times.
The paper introduces Self-Augmented Diffusion Guidance, a physics-informed method that augments diffusion models with self-generated data to enforce physical law constraints. By conditioning the data distribution on the deviation from correct dynamics and setting this deviation to zero during generation, the approach decouples equation evaluation from training and sampling, avoiding costly numerical solves. Experiments show the method markedly reduces deviations versus standard diffusion models and further improves results when combined with existing physics-constrained diffusion techniques.
arXiv:2608. 04827v1 Announce Type: cross Abstract: We introduce the Intrinsic Hybrid Latent Diffusion Model (ILDM), a generative framework that integrates probabilistic dimensionality reduction with geometry-aware diffusion on unknown manifolds.
The paper introduces a physics‑informed diffusion guidance technique that uses self‑generated data augmentation to condition the diffusion model on the deviation from physical laws. By setting this deviation to zero during sampling, the method decouples equation evaluation from training and sampling, eliminating the need to solve governing equations at each denoising step. Experiments show the approach substantially reduces deviations from true dynamics and further improves performance when combined with existing physics‑constrained diffusion methods.
arXiv:2505. 23527v4 Announce Type: replace Abstract: Modern reinforcement learning (RL) algorithms have found success by using powerful probabilistic models, such as transformers, energy-based models, and diffusion/flow-based models.
arXiv:2601. 21284v2 Announce Type: replace-cross Abstract: Diffusion models have emerged as powerful generative tools for modeling complex data distributions, yet their purely data-driven nature limits applicability in engineering and scientific problems where physical laws must be respected.