Diffusion-Probabilistic-Models-Solver

This repository contains an implementation and in-depth mathematical analysis of the paper:

DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps
by Jiaming Song, Chenlin Meng, and Stefano Ermon (NeurIPS 2022).

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About the Project

Diffusion Probabilistic Models (DPMs) are powerful generative models but often require hundreds or thousands of sampling steps to generate high-quality data. The DPM-Solver significantly reduces this to around 10 steps, using a tailored high-order ODE solver.

This repository:

• Re-implements the DPM-Solver-1, DPM-Solver-2, and DPM-Solver-3 methods.
• Includes step-by-step mathematical derivations of the algorithm.
• Reproduces experiments with toy datasets and CIFAR-10, comparing performance with DDPM and DDIM solvers.

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Theoretical Insights

The core idea is to transform the stochastic differential equations of DPMs into a probability flow ODE, taking advantage of its semi-linear structure.

The reverse process relies on a neural network, ϵθ(x, t), to estimate the noise (score function). Using variation of constants, the paper derives an efficient ODE solver with an analytical form of the solution, enabling exponential integration.

We derived and proved:

• The closed-form ODE solution via exponential integrators.
• High-order solver schemes up to 3rd order with error analysis.
• The mathematical foundations and assumptions behind the solver design.

Contents of the Supplementary Document:

• A step-by-step derivation of the forward diffusion SDE and its equivalent form.
• Reverse-time SDE derivation using Anderson's result and substitution of the score function.
• Derivation of the loss function used to train the noise-prediction model ϵθ(x, t).
• Transformation of the reverse SDE into a probability flow ODE and its semi-linear formulation.
• Variation of parameters method applied to derive the general ODE solution in integral form.
• Definition and role of the transformation function λ(t) = log(α(t)/σ(t)).
• Derivation of the solver update rule (Proposition 1) as an exponentially weighted integral.
• Complete derivation of high-order solvers using.

For full derivations and proofs, refer to the supplementary document:
Suplementary text with demonstrations - DPMSolver.pdf