\section{Introduction}




Flow-based generative modeling \citep{flow_matching,rectified_flow,av22,bah+23,ekb+24} has recently gained substantial traction in machine learning due to its capacity to learn expressive, invertible transformations that map simple source distributions to more complex target distributions. In particular, flow matching techniques \citep{flow_matching,rectified_flow} have shown promising results in bridging the gap between traditional normalizing flows and score-based diffusion models. These methods typically construct a continuous time trajectory or ``flow'' that transports samples from a prior distribution, usually Gaussian distribution, to an unknown data distribution. By matching a parameterized velocity field to the ground-truth time derivatives along a path connecting these distributions, flow matching has demonstrated impressive empirical performance, as well as favorable theoretical properties.

Despite these advances, existing flow-based frameworks remain susceptible to perturbations such as noise contamination in the data or instability in the learned transport path \citep{wet+24,hwa+24}. This vulnerability arises because standard (i.e., first-order) methods predominantly focus on velocity alignment, thereby neglecting higher-order dynamics and their influence on smoothness and robustness. Several works in diffusion-based modeling \citep{c23,hg24,llly24} have suggested that carefully accounting for noise and incorporating additional constraints can lead to more stable solutions. However, a principled and comprehensive approach to integrating higher-order information within flow matching has yet to be fully explored.

In this paper, we propose NRFlow, a novel extension to the traditional flow-based generative framework that leverages acceleration fields in addition to velocity fields. Our approach is motivated by the observation that second-order information can be interpreted as a form of regularization, acting to enforce higher-order smoothing constraints on the learned trajectories. Concretely, we show—both formally and informally—that these second-order terms can mitigate noisy or imperfect training data by providing stronger regularity conditions, which in turn bolster model robustness. The core idea is straightforward but powerful: we decompose the learning objective into two parts, one for velocity matching and one for acceleration matching, and jointly train these terms to ensure smooth, stable flows.

Our main theoretical contributions center on establishing a rigorous noise-robustness guarantee that quantifies how noise in the observed data propagates through the learned second-order flow. Specifically, we derive a two-part loss function whose first-order component learns a velocity field approximating $\dot{x}_t$,  while the second-order component targets $\ddot{x}_t$. We then prove that if these losses remain small, the Sobolev $H^2$-norm of the estimation error is bounded, effectively demonstrating the regularization effect of the second-order term. Further, a discrete Gronwall-type analysis reveals that noise in the training distribution propagates sublinearly over time, thereby yielding improved stability compared with purely first-order flow matching.

In addition to the theoretical framework, we also propose a second-order inference algorithm that modifies the classic flow integration step by adding an acceleration update. Experimental results on a Gaussian mixture dataset highlight the effectiveness of our approach.


The summary of our contributions to the theoretical understanding of these architectures and
their boundaries, showed as follows:

\begin{itemize}
    \item We develop a unified second-order flow formulation offering a broad perspective on incorporating higher-order dynamics into generative models.
    \item We establish rigorous guarantees showing that NRFlow exhibits improved immunity to data noise, grounded in both an informal explanation of its regularization properties and a formal theorem bounding noise propagation.
\end{itemize}