Stochastic differential equation

Stochastic differential equations (SDEs) refer to equations that contain randomness in the form of noise terms. In generative modeling, SDEs provide a way to define diffusion processes that gradually add noise to data samples like images. Carefully designed SDEs transform data into predictable noise distributions. The key insight is that this noising process can be reversed by sampling from the distributions to recover the original data.

By manipulating the SDE, researchers can control how noise is inserted over multiple diffusion steps. The sampled data incrementally loses structure and becomes unrecognizable. But reversed through the SDE, high fidelity data can be synthesized. The perturbations guided by the stochastic behavior of the SDE are crucial for the model to generate realistic novel samples.

Models like DDPM and DDIM rely on SDE-defined diffusions that enable training generative neural networks. Architectural optimizations introduced by researchers like Timo Karras further improve the quality by adding components like predictor networks and classifier guidance. The flexibility of SDEs to stochastically model diffusions is what empowers cutting-edge generative models to create remarkable photorealistic images and beyond simply from sampling noise.