ayz2/hamiltonian_pdes

Hamiltonian Neural PDE Solvers through Functional Approximation (NeurIPS 2025)

1D Datasets

1D data is generated for the 1D Advection and 1D KdV equations, from Masked Autoencoder are PDE Learners and Lie Point Symmetry Data Augmentation for Neural PDE Solvers. Adv datasets have a resolution of 128, with 200 timesteps for train and 1000 timesteps for validation. KdV datasets have a resolution of 256, with 200 timesteps for train and 200 timesteps for validation (although during training, only 50 timesteps are used in the paper).

The datasets are organized as:

- dataset.h5 (keys: 'train' or 'valid')
    - 'train' (keys: 't', 'u', 'x')
        - 't': shape (num_samples, num_timesteps). Contains the time at each timestep for each sample.
        - 'u': shape (num_samples, num_timesteps, resolution_x). Contains the data at each nodal position, timestep, and sample. 
        - 'x': shape (num_samples, resolution_x). Contains the x-position for each position along the x-axis for each sample.

A list of 2048 KdV samples with the lowest variation in the Hamiltonian is also provided (H_list_KdV.pkl). This is because the solver is not 100% accurate and the KdV Hamiltonian is highly nonlinear; therefore the ground-truth may not conserve the Hamiltonian even though theoretically it should. Therefore, we can filter some of the worst offenders out during training to avoid learning non-conservative behavior.

2D Datasets

2D Data is generated for the 2D Shallow-Water Equations, from PyClaw. A solver implementation is included in the github repo. SWE datasets have a resolution of (256x256) and 101 timesteps. The SWE_Gaussian dataset is generated only for validation to test OOD behavior after training on sinusoidal initial conditions.

The datasets are organized as:

- dataset.h5 (keys: 'train' or 'valid')
    - 'train' (keys: 't', 'u', 'x')
        - 't': shape (num_timesteps). Contains the time at each timestep for each sample. Assumed to be constant for each sample.
        - 'u': shape (num_samples, num_timesteps, resolution_x, resolution_y, 3). Contains the data (height, velocity_x, velocity_y) at each nodal position, timestep, and sample. 
        - 'x': shape (resolution_x, resolution_y, 2). Contains the x and y coordinates for each sample. Assumed to be constant for each sample.

Normalization statistics calculated on the training dataset are also provided if you want to train with a normalizer, since the height field has a different scale than velocity_x and velocity_y.