Comparison of neural closure models for discretised PDEs

TL;DR

This paper compares different training procedures for neural closure models in discretised PDEs, finding that trajectory fitting with discretise-then-optimise offers the best balance of accuracy and robustness.

Contribution

It systematically evaluates three training procedures for neural closure models, providing insights into their relative performance and long-term accuracy.

Findings

Trajectory fitting is more robust than derivative fitting.

Discretise-then-optimise yields more accurate models than optimise-then-discretise.

Long-term accuracy depends on short-term training accuracy, as explained by new interpretations of existing theorems.

Abstract

Neural closure models have recently been proposed as a method for efficiently approximating small scales in multiscale systems with neural networks. The choice of loss function and associated training procedure has a large effect on the accuracy and stability of the resulting neural closure model. In this work, we systematically compare three distinct procedures: "derivative fitting", "trajectory fitting" with discretise-then-optimise, and "trajectory fitting" with optimise-then-discretise. Derivative fitting is conceptually the simplest and computationally the most efficient approach and is found to perform reasonably well on one of the test problems (Kuramoto-Sivashinsky) but poorly on the other (Burgers). Trajectory fitting is computationally more expensive but is more robust and is therefore the preferred approach. Of the two trajectory fitting procedures, the discretise-then-optimise approach produces more accurate models than the optimise-then-discretise approach. While the optimise-then-discretise approach can still produce accurate models, care must be taken in choosing the length of the trajectories used for training, in order to train the models on long-term behaviour while still producing reasonably accurate gradients during training. Two existing theorems are interpreted in a novel way that gives insight into the long-term accuracy of a neural closure model based on how accurate it is in the short term.