Thank you for reviewing our manuscript.

We recommend using Conda for environment management. You can create a new environment as follows:

conda env create -f environment.yml
conda activate VarRUOT

VarRUOT has been tested on Linux systems with CUDA available and on Windows 11.

All training operations are performed in train.py. First, specify the dataset. For example, if you want to use the three-gene simulation dataset, use:

dataset = SimulationDataset()

Next, define a network to approximate $\lambda(\mathbf{x},t)$ based on the dimensions of the dataset:

multipliermodel = MultiplierModel(data_dim=2)

Create an instance of the VarRUOT class. If you want to use the standard WFR metric, use:

my_ruot = VarRUOT_WFR(dataset, multiplyer_model=multipliermodel)

If you prefer to use the modified metric, use:

my_ruot = VarRUOT_modified(dataset, multiplyer_model=multipliermodel)

Specify the path for saving the model and start training:

train(my_ruot, save_path="mouse_easy")

The model will be saved in the checkpoints folder. To visualize the model's learning trajectory, use:

my_ruot.visualize2dresult()

If you want to see the growth values of the data points, use:

my_ruot.visualizedatagrowth()

Below are the names and dimensions of several datasets:

• Three-Gene Simulation Dataset: SimulationDataset(), 2 dimensions
• EMT Dataset: EmtDataset, 10 dimensions
• Mouse Hematopoiesis (2D): MouseDataset(), 2 dimensions
• Pancreatic $\beta$-Cell Differentiation Dataset: VeresDataset(), 30 dimensions
• Mouse Hematopoiesis (50D): Dim50Dataset(), 50 dimensions
• 50-Dimensional Gaussian Dataset: Gaussian50Dataset(), 50 dimensions
• 100-Dimensional Gaussian Dataset: Gaussian100Dataset(), 100 dimensions

All datasets can be trained using the process described above.