https://www.chenk.top/en/tags/sde/Recent content in SDE on Chen Kai BlogHugoenTue, 30 Jul 2024 09:00:00 +0000https://www.chenk.top/en/pde-ml/07-diffusion-models/Tue, 30 Jul 2024 09:00:00 +0000https://www.chenk.top/en/pde-ml/07-diffusion-models/<p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/pde-ml/07-Diffusion-Models/illustration_1.png" alt="PDE and ML (7): Diffusion Models and Score Matching — Chapter overview" loading="lazy" decoding="async" class="content-image"> </figure> </p> <hr> <p>The output side of a diffusion model is familiar: a high-quality image. The training objective, on the other hand, looks counter-intuitive at first sight — <strong>add noise to the data until it is fully Gaussian, then learn to denoise step by step</strong>. Why is this detour more effective than learning the data distribution directly?</p> <p>The answer is hidden in PDEs. The forward noising process is a <strong>heat equation</strong> (or, more generally, a Fokker–Planck equation), and it admits a reverse-time version — provided we know the score (the gradient of the log-density) at every time. <strong>Score matching</strong> is the standard way to learn that score. From this angle, DDPM, DDIM, and score-based SDEs are not three different algorithms but three discretizations of the same PDE story.</p>