# I-SplineFlow: Learning Monotone Spline Stochastic Interpolant Schedulers for Few-Step Generation

arXiv: https://arxiv.org/abs/2609.27963 · HTML: https://mlsanity.com/paper/2609.27963/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 19 references that are in the map.
- First posted on arXiv 23 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Training-free Linear Image Inverses via Flows](https://mlsanity.com/paper/2310.04432/) (2023)
- [Accelerating Diffusion Sampling with Optimized Time Steps](https://mlsanity.com/paper/2402.17376/) (2024)
- [On the Trajectory Regularity of ODE-based Diffusion Sampling](https://mlsanity.com/paper/2405.11326/) (2024)
- [Learning to Discretize Denoising Diffusion ODEs](https://mlsanity.com/paper/2405.15506/) (2024)
- [Bespoke Solvers for Generative Flow Models](https://mlsanity.com/paper/2310.19075/) (2023)
- [FlowDCN: Exploring DCN-like Architectures for Fast Image Generation with Arbitrary Resolution](https://mlsanity.com/paper/2410.22655/) (2024)
- [B'ezierFlow: Learning B'ezier Stochastic Interpolant Schedulers for Few-Step Generation](https://mlsanity.com/paper/2512.13255/) (2025)
- [DyWeight: Dynamic Gradient Weighting for Few-Step Diffusion Sampling](https://mlsanity.com/paper/2603.11607/) (2026)

## Abstract (arXiv)

Few-step generation with pretrained diffusion and flow models can be accelerated by lightweight training that optimizes the sampling trajectory rather than the network. A recent approach parameterizes the stochastic interpolant (SI) scheduler as a smooth curve whose control points enforce the three properties an SI scheduler must satisfy: fixed boundary conditions, a monotone signal-to-noise ratio (SNR), and differentiability. Existing parameterizations use globally supported polynomial bases, where every control point moves the whole curve and higher expressiveness needs a higher degree, which couples distant regions of the schedule during optimization. We introduce I-SplineFlow, which parameterizes the scheduler with integrated monotone splines (I-splines). I-splines decouple the polynomial degree from the number of mixture weights, so support width and smoothness can be chosen per model at a fixed weight count, and the compactly supported derivative basis makes the scheduler Jacobian orders of magnitude better conditioned than a B'ezier basis. Boundary conditions and a strictly monotone SNR hold by construction, with no ordering constraint on the parameters and closed-form velocity derivatives. Across diffusion (EDM) and flow (ReFlow, Simple ReFlow) models, I-SplineFlow improves few-step FID over B'ezier scheduling in most settings, most clearly at the lowest NFEs, and trains in minutes. Ablations show that both the degree freedom and the monotonicity constraint are needed. The code will be released upon acceptance.