  A potent class of generative models known as Diffusion Probabilistic Models
(DPMs) has become prominent. A forward diffusion process adds gradually noise
to data, while a model learns to gradually denoise. Sampling from pre-trained
DPMs is obtained by solving differential equations (DE) defined by the learnt
model, a process which has shown to be prohibitively slow. Numerous efforts on
speeding-up this process have consisted on crafting powerful ODE solvers.
Despite being quick, such solvers do not usually reach the optimal quality
achieved by available slow SDE solvers. Our goal is to propose SDE solvers that
reach optimal quality without requiring several hundreds or thousands of NFEs
to achieve that goal. We propose Stochastic Explicit Exponential
Derivative-free Solvers (SEEDS), improving and generalizing Exponential
Integrator approaches to the stochastic case on several frameworks. After
carefully analyzing the formulation of exact solutions of diffusion SDEs, we
craft SEEDS to analytically compute the linear part of such solutions. Inspired
by the Exponential Time-Differencing method, SEEDS use a novel treatment of the
stochastic components of solutions, enabling the analytical computation of
their variance, and contains high-order terms allowing to reach optimal quality
sampling $\sim3$-$5\times$ faster than previous SDE methods. We validate our
approach on several image generation benchmarks, showing that SEEDS outperform
or are competitive with previous SDE solvers. Contrary to the latter, SEEDS are
derivative and training free, and we fully prove strong convergence guarantees
for them.
