PODTrajectoryModule#

class PODTrajectoryModule(time_grid, trajectories, n_modes, poly_order, additional_data=None, batch_size=None, is_periodic=False)[source]#

Captures the trajectories of an arbitrary set of particles (e.g. Nodes of structured meshes) by computing the proper orthogonal decomposition (POD) from a set of snapshots.

\[u(t) = \Sigma_j^{N_{modes}} \phi_j w_j(t),\]

where \(\phi_j\) are the computed basis functions (modes) $w_j(t)$ are the corresponding mode-weights as function of time. To allow the motion state to be reconstructed from the low-rank representation at arbitrary time-points within the interval of specified snapshots, the mode-weights are represented as Taylor-series.

Example Usage:
data, t = ...  # get snapshots of states and corresponding time-points
# Shapes: data.shape == (#particles, #snapshots, #channels)
#         t.shape == (#steps, )

pod_module = cmrsim.trajectory.PODTrajectoryModule(data, t, n_modes=5, poly_oder=8)

new_time_grid = np.linspace(t[0].m, t[-1].m, 100).astype(np.float32)

reconstructed_states, _ = pod_module(new_time_grid)
## reconstructed_states.shape == (#particles, 100, #channels)
Parameters:

  • time_grid (ndarray) – (#time_steps)

  • trajectories (ndarray) – (#particles, #time_steps, 3)

  • n_modes (int) – Number of modes used for reduce-order representation

  • poly_order (int) – Order of the Taylor-series used to fit the mode-weights

  • additional_data (Dict[str, ndarray]) – Dict[str, np.ndarray] of shape (#particles, #time_steps, #channels)

  • batch_size (int) – int

  • is_periodic (bool) –

Methods:

__call__(initial_positions, timing[, ...])

Reconstructs the data state at given times t, by evaluating the taylor series of mode-weights and computing the weighted sum.

calculate_pod(time_grid, data, n_modes[, ...])

Computes the proper orthogonal decomposition of data snapshots at points defined in time_grid.

increment_particles(particle_positions, dt, ...)

Evaluates the particle position for the time self.current_time_ms + dt and adds the delta t to the current_time_ms variable

Attributes:

basis_function

Computed basis-functions (modes) \(\phi_j\) used to represent the input data in a reduced order.

batch_size

Together with self.current_batch_size determines the subset of particle trajectories that is evaluated on call and increment_particles

current_batch_idx

Allows to only evaluate the position for a batch of stored particle trajectories

current_time_ms

Keeps track of the current timing when increment_particles is called.

end_time

n_modes

Number of modes used for reduce-order representation

ref_time

__call__(initial_positions, timing, batch_index=0, **kwargs)[source]#

Reconstructs the data state at given times t, by evaluating the taylor series of mode-weights and computing the weighted sum.

Parameters:

  • timing (Tensor) – (#timesteps) in milliseconds

  • initial_positions (Tensor) –

  • batch_index (int) –

Return type:

(Tensor, dict)

Returns:

(#particles, #timesteps, self._channels), {k: v} - additional data

static calculate_pod(time_grid, data, n_modes, remove_mean=False)[source]#

Computes the proper orthogonal decomposition of data snapshots at points defined in time_grid. Returns only the n_modes number of most significant modes

Parameters:

  • time_grid (ndarray) – (#time_steps) time-points corresponding to snapshots

  • data (ndarray) – (#particles, #time_steps, #channels) snapshots of data

  • n_modes (int) – number of most significant modes to return

  • remove_mean (bool) – if true removes the temporal mean of all snapshots before computing POD

Return type:

(ndarray, ndarray)

Returns:

  • POD base modes, shape: (#particles * #channels, n_modes),

  • scaling of modes per time-step (#time_steps, n_modes)

increment_particles(particle_positions: ~tensorflow.python.framework.ops.Tensor, dt: ~tensorflow.python.framework.ops.Tensor, **kwargs) -> (<class 'tensorflow.python.framework.ops.Tensor'>, <class 'dict'>)[source]#

Evaluates the particle position for the time self.current_time_ms + dt and adds the delta t to the current_time_ms variable

Parameters:

  • particle_positions (Tensor) – unused parameter (to adhere to calling signature of trajectory modules)

  • dt (Tensor) – temporal step lengths

  • kwargs – unused parameter (to adhere to calling signature of trajectory modules)

Return type:

(Tensor, dict)

Returns:

(#batch, self._channels), {k: v} - additional data

basis_function: Variable#

Computed basis-functions (modes) \(\phi_j\) used to represent the input data in a reduced order. Shape (#particles * #channels, #modes)

batch_size: Variable#

Together with self.current_batch_size determines the subset of particle trajectories that is evaluated on call and increment_particles

current_batch_idx: Variable#

Allows to only evaluate the position for a batch of stored particle trajectories

current_time_ms: Variable#

Keeps track of the current timing when increment_particles is called.

end_time: Variable#
n_modes: Variable#

Number of modes used for reduce-order representation

ref_time: Variable#