Machine learning algorithms for three-dimensional mean-curvature computation in the level-set method

Abstract

We propose a data-driven mean-curvature solver for the level-set method. This work is the natural extension to R3 of our two-dimensional error-correcting strategy presented in Larios-Cárdenas and Gibou (October 2022) [1] and the hybrid inference system of Larios-Cárdenas and Gibou (August 2022) [2]. However, in contrast to [1,2], which built resolution-dependent dictionaries of curvature neural networks, here we develop a single pair of neural models in R3, regardless of the mesh size. The core of our system comprises two feedforward neural networks. These models ingest preprocessed level-set, gradient, and curvature information to fix numerical mean-curvature approximations for selected nodes along the interface. To reduce the problem's complexity, we have used the local Gaussian curvature κG to classify stencils and fit these networks separately to non-saddle (where κG⪆0) and saddle (where κG⪅0) input patterns. The Gaussian curvature is essential for enhancing generalization and simplifying neural network design. For example, non-saddle stencils are easier to handle because they exhibit a mean-curvature error distribution characterized by monotonicity and symmetry. While the latter has allowed us to train only on half the mean-curvature spectrum, the former has helped us blend the machine-learning-corrected output and the baseline estimation seamlessly around near-flat regions. 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 stencil distinction makes learning-tuple generation in R3 quite involving. 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 data sets is systematic but gleans samples randomly while ensuring well-balancedness. Furthermore, 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 to improve stability and precision at inference time. The synergy among all these features has led to a performant mean-curvature solver that works for any grid resolution. Experiments with several interfaces confirm that our proposed system can yield more accurate mean-curvature estimations in the L1 and L∞ norms than modern particle-based interface reconstruction and level-set schemes around under-resolved regions. Although there is ample room for improvement, this research already shows the potential of machine learning to remedy geometrical problems in well-established numerical methods. Our neural networks are available online at https://github.com/UCSB-CASL/Curvature_ECNet_3D.