# A hybrid inference system for improved curvature estimation in the level-set method using machine learning

arXiv: https://arxiv.org/abs/2104.02951 · HTML: https://mlsanity.com/paper/2104.02951/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 87 references that are in the map.
- First posted on arXiv 7 Apr 2021.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Deep contextualized word representations](https://mlsanity.com/paper/1802.05365/) (2018)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Enriching Word Vectors with Subword Information](https://mlsanity.com/paper/1607.04606/) (2016)
- [To prune, or not to prune: exploring the efficacy of pruning for model compression](https://mlsanity.com/paper/1710.01878/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [A high-bias, low-variance introduction to Machine Learning for physicists](https://mlsanity.com/paper/1803.08823/) (2018)
- [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)

## Built on by (most cited first, 3 of 3)

- [Error-correcting neural networks for semi-Lagrangian advection in the level-set method](https://mlsanity.com/paper/2110.11611/) (2021)
- [Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-Set Method](https://mlsanity.com/paper/2201.12342/) (2022)
- [Machine learning algorithms for three-dimensional mean-curvature computation in the level-set method](https://mlsanity.com/paper/2208.09047/) (2022)

## Abstract (arXiv)

We present a novel hybrid strategy based on machine learning to improve curvature estimation in the level-set method. The proposed inference system couples enhanced neural networks with standard numerical schemes to compute curvature more accurately. The core of our hybrid framework is a switching mechanism that relies on well established numerical techniques to gauge curvature. If the curvature magnitude is larger than a resolution-dependent threshold, it uses a neural network to yield a better approximation. Our networks are multilayer perceptrons fitted to synthetic data sets composed of sinusoidal- and circular-interface samples at various configurations. To reduce data set size and training complexity, we leverage the problem's characteristic symmetry and build our models on just half of the curvature spectrum. These savings lead to a powerful inference system able to outperform any of its numerical or neural component alone. Experiments with stationary, smooth interfaces show that our hybrid solver is notably superior to conventional numerical methods in coarse grids and along steep interface regions. Compared to prior research, we have observed outstanding gains in precision after training the regression model with data pairs from more than a single interface type and transforming data with specialized input preprocessing. In particular, our findings confirm that machine learning is a promising venue for reducing or removing mass loss in the level-set method.