# Error-correcting neural networks for semi-Lagrangian advection in the level-set method

arXiv: https://arxiv.org/abs/2110.11611 · HTML: https://mlsanity.com/paper/2110.11611/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 108 references that are in the map.
- First posted on arXiv 22 Oct 2021.

## Builds on (most cited first, 21 of 21)

- [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)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Enriching Word Vectors with Subword Information](https://mlsanity.com/paper/1607.04606/) (2016)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (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)
- [tempoGAN: A Temporally Coherent, Volumetric GAN for Super-resolution Fluid Flow](https://mlsanity.com/paper/1801.09710/) (2018)
- [Solving inverse-PDE problems with physics-aware neural networks](https://mlsanity.com/paper/2001.03608/) (2020)
- [Deep Learning the Physics of Transport Phenomena](https://mlsanity.com/paper/1709.02432/) (2017)
- [Using Machine Learning to Augment Coarse-Grid Computational Fluid Dynamics Simulations](https://mlsanity.com/paper/2010.00072/) (2020)
- [A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems](https://mlsanity.com/paper/2106.05587/) (2021)
- [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)

## Built on by (most cited first, 2 of 2)

- [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 machine learning framework that blends image super-resolution technologies with passive, scalar transport in the level-set method. Here, we investigate whether we can compute on-the-fly, data-driven corrections to minimize numerical viscosity in the coarse-mesh evolution of an interface. The proposed system's starting point is the semi-Lagrangian formulation. And, to reduce numerical dissipation, we introduce an error-quantifying multilayer perceptron. The role of this neural network is to improve the numerically estimated surface trajectory. To do so, it processes localized level-set, velocity, and positional data in a single time frame for select vertices near the moving front. Our main contribution is thus a novel machine-learning-augmented transport algorithm that operates alongside selective redistancing and alternates with conventional advection to keep the adjusted interface trajectory smooth. Consequently, our procedure is more efficient than full-scan convolutional-based applications because it concentrates computational effort only around the free boundary. Also, we show through various tests that our strategy is effective at counteracting both numerical diffusion and mass loss. In simple advection problems, for example, our method can achieve the same precision as the baseline scheme at twice the resolution but at a fraction of the cost. Similarly, our hybrid technique can produce feasible solidification fronts for crystallization processes. On the other hand, tangential shear flows and highly deforming simulations can precipitate bias artifacts and inference deterioration. Likewise, stringent design velocity constraints can limit our solver's application to problems involving rapid interface changes. In the latter cases, we have identified several opportunities to enhance robustness without forgoing our approach's basic concept.