Flow-based generative models have shown promise in various machine learning applications, but they often face challenges in handling noise and ensuring robustness in trajectory estimation. In this work, we propose NRFlow, a novel extension to flow-based generative modeling that incorporates second-order dynamics through acceleration fields. We develop a comprehensive theoretical framework to analyze the regularization effects of high-order terms and derive noise robustness guarantees. Our method leverages a two-part loss function to simultaneously train first-order velocity fields and high-order acceleration fields, enhancing both smoothness and stability in learned transport trajectories. These results highlight the potential of high-order flow matching for robust generative modeling in complex and noisy environments.
 