Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-Set Method

Machine Learning Algorithms for Three-Dimensional Mean-Curvature Computation in the Level-Set Method

“…Compared to a network-only approach [48] and an earlier hybrid inference system [2], error correction has proven more effective at closing the gap between the expected curvature and its numerical estimation. Indeed, beyond-usual contextual information (e.g., level-set values, gradients, and the curvature itself), dimensionality reduction, and regularization have been important factors in achieving the results reported in [1].…”

“…In this manuscript, we propose to address the level-set's curvature shortcomings from a data-driven perspective. The latest developments of Qi et al [46] and Patel et al [47], in particular, have inspired our machine-and deep-learning incursions for solving not only for mean-curvature [1,2,48] but also passive transport [49]. In the pioneering study of Qi et al [46], the researchers fitted two-layered perceptrons to circular-interface samples to estimate curvature from area fractions in the VOF representation.…”

“…Our research incorporates the notion of neural error quantification first introduced in computational fluid dynamics to fix artificial numerical diffusion and augment coarse-grid simulations [44,49]. More importantly, our study is the natural extension to R 3 of the preliminary two-dimensional error-correcting strategy presented in [1]. In previous work, we optimized simple multilayer perceptrons (MLP) [54] with samples from circular and sinusoidal zero-isocontours to offset numerical curvature errors at the interface.…”

“…On the other hand, the saddle-pattern error structure is less clear; thus, we have exploited no latent characteristics beyond what is known. Such a distinction between stencils makes learning-tuple generation in R 3 more involving than its planar counterpart [1]. In this regard, we have trained our models on not only spherical but also sinusoidal and hyperbolic paraboloidal patches at various configurations.…”

“…Our approach to building their corresponding data sets is systematic but gleans samples in a randomized fashion while ensuring well-balancedness as much as possible. In the same spirit as [1,49], we have resorted to standardization and dimensionality reduction as a preprocessing step and integrated layer-wise regularization to minimize outlying effects. In addition, our strategy leverages mean-curvature rotation and reflection invariance, as in [53], to improve stability and precision at inference time.…”