More Details

Background

After the discretization of space and time, the simulation of a time-dependent partial differential equation amounts to the repeated application of a simulation operator \(\mathcal{P}_h\). Here, we are interested in imitating/emulating this physical/numerical operator with a neural network \(f_\theta\). This repository is concerned with an abstract implementation of all ways we can frame a learning problem to inject "knowledge" from \(\mathcal{P}_h\) into \(f_\theta\).

Assume we have a distribution of initial conditions \(\mathcal{Q}\) from which we sample \(S\) initial conditions, \(u^{[0]} \propto \mathcal{Q}\). Then, we can save them in an array of shape \((S, C, *N)\) (with C channels and an arbitrary number of spatial axes of dimension N) and repeatedly apply \(\mathcal{P}\) to obtain the training trajectory of shape \((S, T+1, C, *N)\).

For a one-step supervised learning task, we substack the training trajectory into windows of size \(2\) and merge the two leftover batch axes to get a data array of shape \((S \cdot T, 2, N)\) that can be used in supervised learning scenario

\[ L(\theta) = \mathbb{E}_{(u^{[0]}, u^{[1]}) \sim \mathcal{Q}} \left[ l\left( f_\theta(u^{[0]}), u^{[1]} \right) \right] \]

where \(l\) is a time-level loss. In the easiest case \(l = \text{MSE}\).

Trainax supports way more than just one-step supervised learning, e.g., to train with unrolled steps, to include the reference simulator \(\mathcal{P}_h\) in training, train on residuum conditions instead of resolved reference states, cut and modify the gradient flow, etc.

A Taxonomy of Training Methodologies

The major axes that need to be chosen are:

• The unrolled length (how often the network is applied autoregressively on the input)
• The branch length (how long the reference goes alongside the network; we get full supervised if that is as long as the rollout length)
• Whether the physics is resolved (diverted-chain and supervised) or only given as a condition (residuum-based loss)

Additional axes are:

• The time level loss (how two states are compared, or a residuum state is reduced)
• The time level weights (if there is network rollout, shall states further away from the initial condition be weighted differently (like exponential discounting in reinforcement learning))
• If the main chain of network rollout is interleaved with a physics solver (-> mix chain)
• Modifications to the gradient flow:
  • Cutting the backpropagation through time in the main chain (after each step, or sparse)
  • Cutting the diverted physics
  • Cutting the one or both levels of the inputs to a residuum function.

Implementation details

There are three levels of hierarchy:

1. The loss submodule defines time-level wise comparisons between two states. A state is either a tensor of shape (num_channels, ...) (with ellipsis indicating an arbitrary number of spatial dim,ensions) or a tensor of shape (num_batches, num_channels, ...). The time level loss is implemented for the former but allows additional vectorized and (mean-)aggregated on the latter. (In the schematic above, the time-level loss is the green circle).
2. The configuration submodule devises how neural time stepper \(f_\theta\) (denoted NN in the schematic) interplays with the numerical simulator \(\mathcal{P}_h\). Similar to the time-level loss this is a callable PyTree which requires during calling the neural stepper and some data. What this data contains depends on the concrete configuration. For supervised rollout training it is the batch of (sub-) trajectories to be considered. Other configurations might also require the reference stepper or a two consecutive time level based residuum function. Each configuration is essentially an abstract implementation of the major methodologies (supervised, diverted-chain, mix-chain, residuum). The most general diverted chain implementation contains supervised and branch-one diverted chain as special cases. All configurations allow setting additional constructor arguments to, e.g., cut the backpropagation through time (sparsely) or to supply time-level weightings (for example to exponentially discount contributions over long rollouts).
3. The training submodule combines a configuration together with stochastic minibatching on a set of reference trajectories. For each configuration, there is a corresponding trainer that essentially is sugarcoating around combining the relevant configuration with the GeneralTrainer and a trajectory substacker.

You can find an overview of predictor learning setups here.