# Parametric Generative Schemes with Geometric Constraints for Encoding and Synthesizing Airfoils

arXiv: https://arxiv.org/abs/2205.02458 · HTML: https://mlsanity.com/paper/2205.02458/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 40 references that are in the map.
- First posted on arXiv 5 May 2022.

## Builds on (most cited first, 8 of 8)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [Machine Learning in Aerodynamic Shape Optimization](https://mlsanity.com/paper/2202.07141/) (2022)
- [Airfoil Design Parameterization and Optimization using B'ezier Generative Adversarial Networks](https://mlsanity.com/paper/2006.12496/) (2020)
- [Airfoil GAN: Encoding and Synthesizing Airfoils for Aerodynamic Shape Optimization](https://mlsanity.com/paper/2101.04757/) (2021)
- [Inverse airfoil design method for generating varieties of smooth airfoils using conditional WGAN-gp](https://mlsanity.com/paper/2110.00212/) (2021)

## Built on by (most cited first, 6 of 6)

- [Airfoil Diffusion: Denoising Diffusion Model For Conditional Airfoil Generation](https://mlsanity.com/paper/2408.15898/) (2024)
- [Physics-Assisted Reduced-Order Modeling for Identifying Dominant Features of Transonic Buffet](https://mlsanity.com/paper/2305.13644/) (2023)
- [Generative method for aerodynamic optimization based on classifier-free guided denoising diffusion probabilistic model](https://mlsanity.com/paper/2503.07056/) (2025)
- [FuncGenFoil: Airfoil Generation and Editing Model in Function Space](https://mlsanity.com/paper/2502.10712/) (2025)
- [Airfoil optimization using Design-by-Morphing with minimized design-space dimensionality](https://mlsanity.com/paper/2510.16020/) (2025)
- [Towards a Foundation-Model Paradigm for Aerodynamic Prediction in Three-dimensional Design](https://mlsanity.com/paper/2604.18062/) (2026)

## Abstract (arXiv)

The modern aerodynamic optimization has a strong demand for parametric methods with high levels of intuitiveness, flexibility, and representative accuracy, which cannot be fully achieved through traditional airfoil parametric techniques. In this paper, two deep learning-based generative schemes are proposed to effectively capture the complexity of the design space while satisfying specific constraints. 1. Soft-constrained scheme: a Conditional Variational Autoencoder (CVAE)-based model to train geometric constraints as part of the network directly. 2. Hard-constrained scheme: a VAE-based model to generate diverse airfoils and an FFD-based technique to project the generated airfoils onto the given constraints. According to the statistical results, the reconstructed airfoils are both accurate and smooth, without any need for additional filters. The soft-constrained scheme generates airfoils that exhibit slight deviations from the expected geometric constraints, yet still converge to the reference airfoil in both geometry space and objective space with some degree of distribution bias. In contrast, the hard-constrained scheme produces airfoils with a wider range of geometric diversity while strictly adhering to the geometric constraints. The corresponding distribution in the objective space is also more diverse, with isotropic uniformity around the reference point and no significant bias. These proposed airfoil parametric methods can break through the boundaries of training data in the objective space, providing higher quality samples for random sampling and improving the efficiency of optimization design.