Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-Set Method. (arXiv:2201.12342v1 [math.NA])

We present an error-neural-modeling-based strategy for approximating
two-dimensional curvature in the level-set method. Our main contribution is a
redesigned hybrid solver (Larios-C'{a}rdenas and Gibou (2021)[1]) that relies
on numerical schemes to enable machine-learning operations on demand. In
particular, our routine features double predicting to harness curvature
symmetry invariance in favor of precision and stability. As in [1], the core of
this solver is a multilayer perceptron trained on circular- and
sinusoidal-interface samples. Its role is to quantify the error in numerical
curvature approximations and emit corrected estimates for select grid vertices
along the free boundary. These corrections arise in response to preprocessed
context level-set, curvature, and gradient data. To promote neural capacity, we
have adopted sample negative-curvature normalization, reorientation, and
reflection-based augmentation. In the same manner, our system incorporates
dimensionality reduction, well-balancedness, and regularization to minimize
outlying effects. Our training approach is likewise scalable across mesh sizes.
For this purpose, we have introduced dimensionless parametrization and
probabilistic subsampling during data production. Together, all these elements
have improved the accuracy and efficiency of curvature calculations around
under-resolved regions. In most experiments, our strategy has outperformed the
numerical baseline at twice the number of redistancing steps while requiring
only a fraction of the cost.



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