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.

Submitted to SIAM Journal on Scientific Computing.

In this work, we provide a solution to the disambiguation task by leveraging the traditional techniques of candidate mapping entity generation and local evaluation with some of the latest developments, such as word embeddings. We also consider a graph-based collective process to establish a topical relatedness metric that helps true mapping entities in a document to disambiguate one another through personalized PageRank. The final mapping entities for the given surface forms are obtained by heuristically reincorporating the candidates' local features with their resulting graph score and performing a maximal discriminant selection. The proposed methodology is capable of reaching up to 80% accuracy when it is evaluated against a well known dataset with around 18,000 named entity mentions.