A deep learning approach for the computation of curvature in the level-set method
This work addresses computational difficulties in the level-set method for fluid dynamics or image processing, but it is incremental as it applies existing deep learning techniques to a specific numerical task.
The authors tackled the problem of estimating mean curvature in the level-set method by proposing a deep learning approach using neural networks trained on synthetic data, achieving competitive accuracy with traditional numerical schemes in L1 and L2 norms, particularly in coarse resolutions and steep curvature regions.
We propose a deep learning strategy to estimate the mean curvature of two-dimensional implicit interfaces in the level-set method. Our approach is based on fitting feed-forward neural networks to synthetic data sets constructed from circular interfaces immersed in uniform grids of various resolutions. These multilayer perceptrons process the level-set values from mesh points next to the free boundary and output the dimensionless curvature at their closest locations on the interface. Accuracy analyses involving irregular interfaces, in both uniform and adaptive grids, show that our models are competitive with traditional numerical schemes in the $L^1$ and $L^2$ norms. In particular, our neural networks approximate curvature with comparable precision in coarse resolutions, when the interface features steep curvature regions, and when the number of iterations to reinitialize the level-set function is small. Although the conventional numerical approach is more robust than our framework, our results have unveiled the potential of machine learning for dealing with computational tasks where the level-set method is known to experience difficulties. We also establish that an application-dependent map of local resolutions to neural models can be devised to estimate mean curvature more effectively than a universal neural network.