NeurIPS 2025

How to build a consistency model:
Learning flow maps via self-distillation
Nicholas M. Boffi
Carnegie Mellon University
Michael S. Albergo
Harvard University
Eric Vanden-Eijnden
Courant Institute of Mathematical Sciences
Abstract
Flow-based generative models achieve state-of-the-art sample quality, but require
the expensive solution of a differential equation at inference time. Flow map mod-
els, commonly known as consistency models, encompass many recent efforts to
improve inference-time efficiency by learning the solution operator of this differ-
ential equation. Yet despite their promise, these models lack a unified description
that clearly explains how to learn them efficiently in practice. Here, building on
the methodology proposed in Boffi et al. (2024), we present a systematic algo-
rithmic framework for directly learning the flow map associated with a flow or
diffusion model. By exploiting a relationship between the velocity field underlying
a continuous-time flow and the instantaneous rate of change of the flow map, we
show how to convert any distillation scheme into a direct training algorithm via
self-distillation, eliminating the need for pre-trained teachers. We introduce three
algorithmic families based on different mathematical characterizations of the flow
map: Eulerian, Lagrangian, and Progressive methods, which we show encompass
and extend all known distillation and direct training schemes for consistency mod-
els. We find that the novel class of Lagrangian methods, which avoid both spatial
derivatives and bootstrapping from small steps by design, achieve significantly
more stable training and higher performance than more standard Eulerian and Pro-
gressive schemes. Our methodology unifies existing training schemes under a single
common framework and reveals new design principles for accelerated generative
modeling. Associated code is available at https://github.com/nmboffi/flow-maps.
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Introduction
Generative models based on dynamical systems, suc...