SoF: Second-Order Flow Matching with Learning Acceleration

Continuous-time generative models are bottlenecked by the number of function evaluations (NFE). We introduce Second-Order Flow Matching (SoF), which equips flow-based models with an explicit acceleration field via the squared-generator identity ( mathcalH_t = mathcalG_t^2 + partial_t mathcalG_t ) and a phase-space equivalence, realized by a dual-head backbone enabling single-NFE second-order sampling at ( < 2 % ) overhead. Combined with a schedule-matched sampling formula and an endpoint-refined inference grid, second-order sampling with the trained acceleration head yields large gains over same-model first-order solvers: 50-NFE FID drops from 21.24 to 2.48 on CIFAR-10 and from 32.03 to 3.40 on CelebA-HQ with 2nd-order Heun, reaching 2.41 at 100 NFE on CIFAR-10 without distillation, while DPM-Solver++ (2V) halves the 20-NFE FID relative to the best first-order solver (3.31 versus 7.16). At very low NFE ( (N leq 4 )), first-order solvers remain competitive. Inpainting on CelebA-HQ and LSUN-Church confirms the benefit extends to conditional generation. Code will be released upon acceptance.