dpm-solver
**DPM-Solver** is a family of high-order ODE solvers specifically designed for the probability flow ODE of diffusion models, providing faster and more accurate sampling than generic solvers (Euler, Heun) by exploiting the semi-linear structure of the diffusion ODE. DPM-Solver achieves high-quality generation in 10-20 steps by using exact solutions of the linear component combined with Taylor expansions of the nonlinear (neural network) component.
**Why DPM-Solver Matters in AI/ML:**
DPM-Solver provides the **fastest high-quality sampling** for pre-trained diffusion models without any additional training, distillation, or model modification, making it the default fast sampler for production diffusion model deployments.
• **Semi-linear ODE structure** — The diffusion probability flow ODE dx/dt = f(t)·x + g(t)·ε_θ(x,t) has a linear component f(t)·x (analytically solvable) and a nonlinear component g(t)·ε_θ (requires neural network evaluation); DPM-Solver solves the linear part exactly and approximates the nonlinear part efficiently
• **Change of variables** — DPM-Solver performs the change of variable from x_t to x_t/α_t (scaled prediction), simplifying the ODE to a form where the linear component is eliminated and only the nonlinear ε_θ term requires approximation
• **Multi-step methods** — DPM-Solver-2 and DPM-Solver-3 use previous model evaluations to construct higher-order approximations (analogous to Adams-Bashforth methods), achieving 2nd and 3rd order accuracy with minimal additional computation
• **DPM-Solver++** — An improved variant that uses the data-prediction (x₀-prediction) formulation instead of noise-prediction, providing more stable high-order updates especially for guided sampling and large classifier-free guidance scales
• **Adaptive step scheduling** — DPM-Solver can use non-uniform time step spacing (more steps at high noise, fewer at low noise) to concentrate computation where the ODE trajectory is most curved, further improving quality per evaluation
| Solver | Order | Steps for Good Quality | NFE (Neural Function Evaluations) |
|--------|-------|----------------------|----------------------------------|
| DDIM (Euler) | 1 | 50-100 | 50-100 |
| DPM-Solver-1 | 1 | 20-50 | 20-50 |
| DPM-Solver-2 | 2 | 15-25 | 15-25 |
| DPM-Solver-3 | 3 | 10-20 | 10-20 |
| DPM-Solver++ (2M) | 2 (multistep) | 10-20 | 10-20 |
| DPM-Solver++ (3M) | 3 (multistep) | 8-15 | 8-15 |
**DPM-Solver is the most efficient training-free sampler for diffusion models, exploiting the mathematical structure of the probability flow ODE to achieve high-quality generation in 10-20 neural function evaluations through exact linear solutions and high-order Taylor approximations, establishing itself as the default fast sampler for deployed diffusion models including Stable Diffusion and DALL-E.**