
\begin{figure*}[!ht]
\centering
\subfloat[3-round spiral dataset]{\includegraphics[width=0.25\textwidth]{3round_spin_dataset.pdf}} 
\subfloat[O1 \cite{rectified_flow}]{\includegraphics[width=0.25\textwidth]{1_1_3round_spin_output.pdf}}
\subfloat[O1+O2 (Ours) ]{\includegraphics[width=0.25\textwidth]{1_12_3round_spin_output.pdf}}
\subfloat[O1+O2+O3 (Ours) ]{\includegraphics[width=0.25\textwidth]{1_123_3round_spin_output.pdf}}
\\
\subfloat[dot-hyperbola dataset]{\includegraphics[width=0.25\textwidth]{dothyperbola_dataset.pdf}}
\subfloat[O1 \cite{rectified_flow}]{\includegraphics[width=0.25\textwidth]{1_1_dothyperbola.pdf}}
\subfloat[O1+O2 (Ours) ]{\includegraphics[width=0.25\textwidth]{1_12_dothyperbola.pdf}}
\subfloat[O1+O2+O3 (Ours) ]{\includegraphics[width=0.25\textwidth]{1_123_dothyperbola.pdf}}
\caption{ 
\textbf{NRFlow on 3-round spiral dataset and dot-hyperbola dataset.} From left to right: the first column shows the 3-round spiral dataset and the dot-hyperbola dataset; the second column shows the results of NRFlow optimized with first-order loss (O1) \cite{rectified_flow}; the third column shows the results of NRFlow optimized by first-order and second-order loss (O1+O2) (Ours); the fourth column are the results of NRFlow optimized by first-order, second-order and third-order loss (O1+O2+O3) (Ours). 
Our high-order NRFlows (third column and fourth column) show great capability in modeling complex distribution. 
Quantitative results are shown in Table~\ref{tab:euclidean_distance_complex_datasets_new}. 
}
\label{fig:mixture_of_gaussian_experiment}
\end{figure*}

\begin{table}[!ht] \centering \caption{ \textbf{Euclidean distance loss across three complex distribution datasets under the new trajectory setting.} Lower values indicate higher accuracy in distribution transfer. The optimal values are highlighted in \textbf{Bold}, and the second-best loss values (second-lowest) are represented by \underline{Underlined} numbers for each dataset (row). }
\label{tab:euclidean_distance_complex_datasets_new}
\begin{tabular}{|l|c|c|c|}
% \toprule
\hline
\textbf{Loss}  & \textbf{Five} & \textbf{3-round} & \textbf{Dot} \\
\textbf{terms}  & \textbf{mode} & \textbf{Spiral} & \textbf{Hyperbola} \\
% \midrule		
\hline
O1 \cite{rectified_flow}             & 1.755 & 17.338 & 18.096 \\
\hline
O1 + O2 (Ours)         & \underline{0.956} & \underline{15.514} & \underline{3.823} \\
\hline
O1 + O2 + O3  (Ours)     & \textbf{0.778} & \textbf{11.866} & \textbf{2.959} \\
\hline
% \bottomrule
\end{tabular}
\end{table}

\section{Experiments}\label{sec:experiment}

This section presents a series of experiments to evaluate the effectiveness of our NRFlow. Our results demonstrate that NRFlow significantly improves distribution generation, with the high-order loss playing a key role in enhancing model performance.
In Section~\ref{sec:exp:experiment_setup}, we provide a detailed explanation for the setup of our experiments.
In Section~\ref{sec:exp:result_analysis}, we provide a comprehensive analysis of the effectiveness of our high-order supervision in our NRFlow model. 


\subsection{Experiment Setup} \label{sec:exp:experiment_setup}
We conduct comprehensive evaluations of NRFlow across diverse data distributions and multiple loss function combinations. Notably, the NRFlow implementation using only first-order loss (denoted as O1) corresponds exactly to the baseline Rectified Flow framework \cite{rectified_flow}. Our proposed extensions consist of two configurations: second-order enhanced (O1+O2) and third-order augmented (O1+O2+O3) variants.

The evaluation employs two challenging synthetic datasets: a three-round spiral distribution and a dot-hyperbola distribution. Each dataset contains 100 sample points drawn from both source and target distributions. Our implementation utilizes a 2-layer fully connected network with 100 hidden units, trained using the Adam optimizer with a learning rate of 0.005. For the three-round spiral dataset, we employ full-batch training (batch size 1000) over 1000 optimization steps. The dot-hyperbola configuration uses an increased batch size of 1600 to account for its greater geometric complexity. Numerical integration is performed using an adaptive ODE solver throughout all experiments.

\subsection{Results Analysis} \label{sec:exp:result_analysis}
Our primary objective involves learning optimal transport trajectories between source distributions (depicted in orange in Figure~\ref{fig:mixture_of_gaussian_experiment}) and target distributions (shown in pick). Empirical results demonstrate that the baseline Rectified Flow model (O1) exhibits significant limitations in target distribution modeling. As visualized in Figure~\ref{fig:mixture_of_gaussian_experiment} (second column), the first-order model generates substantial out-of-distribution artifacts for both synthetic datasets. This observation is quantitatively confirmed in Table~\ref{tab:euclidean_distance_complex_datasets_new}, where O1 has the highest Euclidean distance metrics among all settings.

The introduction of second-order regularization (O1+O2) yields marked improvements.  
Final optimization with third-order constraints (O1+O2+O3) produces the most accurate distribution alignment, achieving near-perfect coverage of target domains. These results conclusively demonstrate that high-order supervision progressively enhances the model's ability to capture complex distributional geometries, with each additional regularization term contributing to statistically significant performance gains.
 
