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

arXiv: https://arxiv.org/abs/2208.09047 · HTML: https://mlsanity.com/paper/2208.09047/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 103 references that are in the map.
- First posted on arXiv 18 Aug 2022.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A high-bias, low-variance introduction to Machine Learning for physicists](https://mlsanity.com/paper/1803.08823/) (2018)
- [Using Machine Learning to Augment Coarse-Grid Computational Fluid Dynamics Simulations](https://mlsanity.com/paper/2010.00072/) (2020)
- [Reformulating Level Sets as Deep Recurrent Neural Network Approach to Semantic Segmentation](https://mlsanity.com/paper/1704.03593/) (2017)
- [A deep learning approach for the computation of curvature in the level-set method](https://mlsanity.com/paper/2002.02804/) (2020)
- [NPLIC: A Machine Learning Approach to Piecewise Linear Interface Construction](https://mlsanity.com/paper/2007.04244/) (2020)
- [Consistent and symmetry preserving data-driven interface reconstruction for the level-set method](https://mlsanity.com/paper/2104.11578/) (2021)
- [A hybrid inference system for improved curvature estimation in the level-set method using machine learning](https://mlsanity.com/paper/2104.02951/) (2021)
- [Error-correcting neural networks for semi-Lagrangian advection in the level-set method](https://mlsanity.com/paper/2110.11611/) (2021)
- [Recursive Algorithms for Distributed Forests of Octrees](https://mlsanity.com/paper/1406.0089/) (2014)
- [Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-Set Method](https://mlsanity.com/paper/2201.12342/) (2022)

## Abstract (arXiv)

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 strategy in [DOI: 10.1007/s10915-022-01952-2][1] and the hybrid inference system of [DOI: 10.1016/j.jcp.2022.111291][2]. However, in contrast to [1,2], which built resolution-dependent neural-network dictionaries, here we develop a pair of models in ℝ3, regardless of the mesh size. Our feedforward networks ingest transformed level-set, gradient, and curvature data to fix numerical mean-curvature approximations selectively for interface nodes. To reduce the problem's complexity, we have used the Gaussian curvature to classify stencils and fit our models separately to non-saddle and saddle patterns. Non-saddle stencils are easier to handle because they exhibit a 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 data-driven and the baseline estimations seamlessly near flat regions. On the other hand, the saddle-pattern error structure is less clear; thus, we have exploited no latent information beyond what is known. In this regard, we have trained our models on not only spherical but also sinusoidal and hyperbolic paraboloidal patches. Our approach to building their data sets is systematic but gleans samples randomly while ensuring well-balancedness. We have also resorted to standardization and dimensionality reduction and integrated regularization to minimize outliers. In addition, we leverage curvature rotation/reflection invariance to improve precision at inference time. Several experiments confirm that our proposed system can yield more accurate mean-curvature estimations than modern particle-based interface reconstruction and level-set schemes around under-resolved regions.