A Deep Learning Approach for the Computation of Curvature in the Level-Set Method

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 $L^2$ and $L^\infty$ 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.