# Neural Network with Local Converging Input (NNLCI) for Supersonic Flow Problems with Unstructured Grids

arXiv: https://arxiv.org/abs/2310.15299 · HTML: https://mlsanity.com/paper/2310.15299/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 23 Oct 2023.

## Builds on (most cited first, 6 of 6)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [GMLS-Nets: A framework for learning from unstructured data](https://mlsanity.com/paper/1909.05371/) (2019)
- [Solving Maxwell's Equation in 2D with Neural Networks with Local Converging Inputs](https://mlsanity.com/paper/2302.02860/) (2023)

## Built on by (most cited first, 1 of 1)

- [A PNP ion channel deep learning solver with local neural network and finite element input data](https://mlsanity.com/paper/2401.17513/) (2024)

## Abstract (arXiv)

In recent years, surrogate models based on deep neural networks (DNN) have been widely used to solve partial differential equations, which were traditionally handled by means of numerical simulations. This kind of surrogate models, however, focuses on global interpolation of the training dataset, and thus requires a large network structure. The process is both time consuming and computationally costly, thereby restricting their use for high-fidelity prediction of complex physical problems. In the present study, we develop a neural network with local converging input (NNLCI) for high-fidelity prediction using unstructured data. The framework utilizes the local domain of dependence with converging coarse solutions as input, which greatly reduces computational resource and training time. As a validation case, the NNLCI method is applied to study inviscid supersonic flows in channels with bumps. Different bump geometries and locations are considered to benchmark the effectiveness and versability of the proposed approach. Detailed flow structures, including shock-wave interactions, are examined systematically.