Snapshot matrix

The snapshot matrix stores solutions of the high-dimensional ODE (obtained from discretizing a PDE). This is then used to construct reduced bases in a data-driven way. So (for a single parameter^[1]) the snapshot matrix takes the following form:

\[M = \left[\begin{array}{c:c:c:c} \hat{u}_1(t_0) & \hat{u}_1(t_1) & \quad\ldots\quad & \hat{u}_1(t_f) \\ \hat{u}_2(t_0) & \hat{u}_2(t_1) & \ldots & \hat{u}_2(t_f) \\ \hat{u}_3(t_0) & \hat{u}_3(t_1) & \ldots & \hat{u}_3(t_f) \\ \ldots & \ldots & \ldots & \ldots \\ \hat{u}_{2N}(t_0) & \hat{u}_{2N}(t_1) & \ldots & \hat{u}_{2N}(t_f) \\ \end{array}\right].\]

In the above example we store a matrix whose first axis is the system dimension (i.e. a state is an element of $\mathbb{R}^{2n}$) and the second dimension gives the time step.

The starting point for using the snapshot matrix as data for a machine learning model is that all the columns of $M$ live on a lower-dimensional solution manifold and we can use techniques such as POD and autoencoders to find this solution manifold. We also note that the second axis of $M$ does not necessarily indicate time but can also represent various parameters (including initial conditions). The second axis in the DataLoader struct is therefore saved in the field n_params.

Snapshot tensor

The snapshot tensor fulfills the same role as the snapshot matrix but has a third axis that describes different initial parameters (such as different initial conditions).

When drawing training samples from the snapshot tensor we also need to specify a sequence length (as an argument to the Batch struct). When sampling a batch from the snapshot tensor we sample over the starting point of the time interval (which is of length seq_length) and the third axis of the tensor (the parameters). The total number of batches in this case is $\lceil\mathtt{(dl.input\_time_steps - batch.seq\_length) * dl.n\_params / batch.batch_size}\rceil$.