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.

, , , , ,
Computers & Mathematics with Applications
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Melchers, H., Crommelin, D., Koren, B., Menkovski, V., & Sanderse, B. (2023). Comparison of neural closure models for discretised PDEs. Computers & Mathematics with Applications, 143, 94–107. doi:10.1016/j.camwa.2023.04.030