Neural SDEs are a class of generative and discriminative models that parameterize both the drift and diffusion of a stochastic differential equation with neural networks — enabling continuous-time latent variable models, continuous normalizing flows with noise, and uncertainty-aware predictions.

Training Neural SDEs

• Variational: Use variational inference with a posterior SDE and prior SDE.
• Score Matching: Train the score function $\nabla log p_t(z)$ for generative modeling.
• Adjoint Method: Backpropagate through the SDE solver using the stochastic adjoint method.
• KL Divergence: The KL between path measures of two SDEs has a tractable form (Girsanov theorem).

Why It Matters

• Diffusion Models: Score-based generative models (DDPM, score matching) can be viewed through the Neural SDE lens.
• Continuous Latent Dynamics: Model continuous-time stochastic processes in latent space (finance, physics).
• Theory + Practice: Neural SDEs connect deep learning to the rich mathematical theory of stochastic processes.

Neural SDEs are deep learning meets stochastic calculus — combining neural network expressiveness with the mathematical framework of stochastic processes.