We propose a deep learning strategy to compute the mean curvature of an implicit level-set representation of an interface. Our approach is based on fitting neural networks to synthetic datasets of pairs of nodal phi values and curvatures obtained from circular interfaces immersed in different uniform resolutions. These neural networks are multilayer perceptrons that ingest sample level-set values of grid points along a free boundary and output the dimensionless curvature at the center vertices of each sampled neighborhood. Evaluations with irregular (smooth and sharp) interfaces, in both uniform and adaptive meshes, show that our deep learning approach is systematically superior to conventional numerical approximation in the L2 and L-Infinity norms. Our methodology is also less sensitive to steep curvatures and approximates them well with samples collected with fewer iterations of the reinitialization equation, often needed to regularize the underlying implicit function. Additionally, we show that an application-dependent map of local resolutions to neural networks can be constructed and employed to estimate interface curvatures more efficiently than using typically expensive numerical schemes while still attaining comparable or higher precision.

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.