Abstract: A potent class of generative models known as Diffusion Probabilistic Models(DPMs) has become prominent. A forward diffusion process adds gradually noiseto data, while a model learns to gradually denoise. Sampling from pre-trainedDPMs is obtained by solving differential equations (DE) defined by the learntmodel, a process which has shown to be prohibitively slow. Numerous efforts onspeeding-up this process have consisted on crafting powerful ODE solvers.Despite being quick, such solvers do not usually reach the optimal qualityachieved by available slow SDE solvers. Our goal is to propose SDE solvers thatreach optimal quality without requiring several hundreds or thousands of NFEsto achieve that goal. We propose Stochastic Explicit ExponentialDerivative-free Solvers (SEEDS), improving and generalizing ExponentialIntegrator approaches to the stochastic case on several frameworks. After carefully analyzing the formulation of exactsolutions of diffusion SDEs, we craft SEEDS to analytically compute the linearpart 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-orderterms allowing to reach optimal quality sampling $\sim3$-$5\times$ faster than previousSDE 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 fullyprove strong convergence guarantees for them.