We present a hybrid strategy based on deep learning to compute mean curvature in the level-set method. The proposed inference system combines a dictionary of improved regression models with standard numerical schemes to estimate curvature more accurately. The core of our framework is a switching mechanism that relies on well-established numerical techniques to gauge curvature. If the curvature magnitude is larger than a resolution-dependent threshold, it uses a neural network to yield a better approximation. Our networks are multi-layer perceptrons fitted to synthetic data sets composed of circular- and sinusoidal-interface samples at various configurations. To reduce data set size and training complexity, we leverage the problem's characteristic symmetry and build our models on just half of the curvature spectrum. These savings result in compact networks able to outperform any of the system's numerical or neural component alone. Experiments with static interfaces show that our hybrid approach is suitable and notoriously superior to conventional numerical methods in under-resolved and steep, concave regions. Compared to prior research, we have observed outstanding gains in precision after including training data pairs from more than a single interface type and other means of input preprocessing. In particular, our findings confirm that machine learning is a promising venue for devising viable solutions to the level-set method's numerical shortcomings.

Submitted.

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, both in uniform and adaptive grids, show that our models are competitive with traditional numerical schemes in the L1 and L2 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.

SIAM J. Sci. Comput., 43(3): A1754-A1779.