Machines that invent. Flow Matching vs. Diffusion: Mastering ODEs and SDEs in Generative Modeling

If you’ve looked at an AI-generated image or video recently, you’ve witnessed a miracle of modern math. For years, the dream was to create machines that don't just recognize patterns, but invent them. We started with pixels, moved to vectors, and now, we’re talking about the "flow" of information itself. Today, we’re tracing the lineage of generative AI—from its probabilistic roots to the high-speed "straight-line" generation of today. We are diving into the Principles of Flow Matching and Diffusion Models. This journey actually starts back in 2013 with pioneers like Diederik Kingma and Max …