We break the mold in flow-based generative modeling by proposing a new model based on the damped wave equation, also known as telegrapher's equation. Similar to the diffusion equation and Brownian motion, there is a Feynman-Kac type relation between the telegrapher's equation and the stochastic Kac process in 1D. The Kac flow evolves stepwise linearly in time, so that the probability flow is Lipschitz continuous in the Wasserstein distance and, in contrast to diffusion flows, the norm of the velocity is globally bounded. Furthermore, the Kac model has the diffusion model as its asymptotic limit. We extend these considerations to a multi-dimensional stochastic process which consists of independent 1D Kac processes in each spatial component. We show that this process gives rise to an absolutely continuous curve in the Wasserstein space and compute the conditional velocity field starting in a Dirac point analytically. Using the framework of flow matching, we train a neural network that approximates the velocity field and use it for sample generation. Our numerical experiments demonstrate the scalability of our approach, and show its advantages over diffusion models.

翻译：我们通过提出一种基于阻尼波动方程（亦称电报方程）的新模型，打破了基于流的生成建模的固有范式。与扩散方程和布朗运动类似，在一维情况下，电报方程与随机Kac过程之间存在一种费曼-卡茨型关系。Kac流在时间上分段线性演化，使得概率流在Wasserstein距离下是Lipschitz连续的，并且与扩散流不同，其速度范数是全局有界的。此外，Kac模型以扩散模型为其渐近极限。我们将这些考量推广到一个多维随机过程，该过程由每个空间分量上独立的1D Kac过程组成。我们证明了该过程在Wasserstein空间中产生一条绝对连续曲线，并解析地计算了从狄拉克点出发的条件速度场。利用流匹配框架，我们训练了一个近似速度场的神经网络，并将其用于样本生成。我们的数值实验证明了该方法的可扩展性，并展示了其相对于扩散模型的优势。