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

Abstract

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-Cárdenas and Gibou, J. Comput. Phys. (May 2022), 10.1016/j.jcp.2022.111291] 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. 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.

References in corpus (7)

• A high-bias, low-variance introduction to Machine Learning for physicists
• Enforcing Analytic Constraints in Neural-Networks Emulating Physical Systems
• Learned discretizations for passive scalar advection in a 2-D turbulent flow
• Computing Curvature for Volume of Fluid Methods using Machine Learning
• A deep learning approach for the computation of curvature in the level-set method
• Consistent and symmetry preserving data-driven interface reconstruction for the level-set method
• A hybrid inference system for improved curvature estimation in the level-set method using machine learning