Efficient differential equation solvers have significantly reduced the sampling time of diffusion models (DMs) while retaining high sampling quality. Among these solvers, exponential integrators (EI) have gained prominence by demonstrating state-of-the-art performance. However, existing high-order EI-based sampling algorithms rely on degenerate EI solvers, resulting in inferior error bounds and reduced accuracy in contrast to the theoretically anticipated results under optimal settings. This situation makes the sampling quality extremely vulnerable to seemingly innocuous design choices such as timestep schedules. For example, an inefficient timestep scheduler might necessitate twice the number of steps to achieve a quality comparable to that obtained through carefully optimized timesteps. To address this issue, we reevaluate the design of high-order differential solvers for DMs. Through a thorough order analysis, we reveal that the degeneration of existing high-order EI solvers can be attributed to the absence of essential order conditions. By reformulating the differential equations in DMs and capitalizing on the theory of exponential integrators, we propose refined EI solvers that fulfill all the order conditions, which we designate as Refined Exponential Solver (RES). Utilizing these improved solvers, RES exhibits more favorable error bounds theoretically and achieves superior sampling efficiency and stability in practical applications. For instance, a simple switch from the single-step DPM-Solver++ to our order-satisfied RES solver when Number of Function Evaluations (NFE) $=9$, results in a reduction of numerical defects by $25.2\%$ and FID improvement of $25.4\%$ (16.77 vs 12.51) on a pre-trained ImageNet diffusion model.

翻译：高效的微分方程求解器显著降低了扩散模型（DMs）的采样时间，同时保持了较高的采样质量。在这些求解器中，指数积分器（EI）因其展现出的最先进性能而备受关注。然而，现有的基于高阶EI的采样算法依赖于退化的EI求解器，导致其误差界劣于理论上在最优设置下的预期结果，且精度降低。这种情况使得采样质量极易受到看似无关紧要的设计选择（如时间步调度）的影响。例如，一个低效的时间步调度器可能需要两倍的步数才能达到精心优化时间步下的质量。为解决这一问题，我们重新评估了DM中高阶微分求解器的设计。通过深入的阶次分析，我们揭示了现有高阶EI求解器的退化可归因于缺少必要的阶次条件。通过重新表述DM中的微分方程并利用指数积分器理论，我们提出了满足所有阶次条件的改进型EI求解器，并将其命名为精细化指数求解器（RES）。利用这些改进的求解器，RES在理论上具有更优的误差界，并在实际应用中实现了更高的采样效率和稳定性。例如，在预训练的ImageNet扩散模型上，当函数评估次数（NFE）$=9$时，仅将单步DPM-Solver++替换为我们满足阶次条件的RES求解器，即可使数值缺陷减少$25.2\%$，FID提升$25.4\%$（从16.77降至12.51）。