# Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows

arXiv: https://arxiv.org/abs/2109.01514 · HTML: https://mlsanity.com/paper/2109.01514/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 69 references that are in the map.
- First posted on arXiv 3 Sep 2021.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [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)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Understanding disentangling in β-VAE](https://mlsanity.com/paper/1804.03599/) (2018)
- [Information Dropout: Learning Optimal Representations Through Noisy Computation](https://mlsanity.com/paper/1611.01353/) (2016)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [The role of artificial intelligence in achieving the Sustainable Development Goals](https://mlsanity.com/paper/1905.00501/) (2019)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Convolutional-network models to predict wall-bounded turbulence from wall quantities](https://mlsanity.com/paper/2006.12483/) (2020)
- [Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows](https://mlsanity.com/paper/2007.00936/) (2020)
- [Interpretable deep-learning models to help achieve the Sustainable Development Goals](https://mlsanity.com/paper/2108.10744/) (2021)

## Built on by (most cited first, 30 of 45)

- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [β-Variational autoencoders and transformers for reduced-order modelling of fluid flows](https://mlsanity.com/paper/2304.03571/) (2023)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [A Denoising Diffusion Model for Fluid Field Prediction](https://mlsanity.com/paper/2301.11661/) (2023)
- [Super-Resolution Analysis via Machine Learning: A Survey for Fluid Flows](https://mlsanity.com/paper/2301.10937/) (2023)
- [Aero-Nef: Neural Fields for Rapid Aircraft Aerodynamics Simulations](https://mlsanity.com/paper/2407.19916/) (2024)
- [Improving aircraft performance using machine learning: a review](https://mlsanity.com/paper/2210.11481/) (2022)
- [Data-driven control of spatiotemporal chaos with reduced-order neural ODE-based models and reinforcement learning](https://mlsanity.com/paper/2205.00579/) (2022)
- [Assessments of epistemic uncertainty using Gaussian stochastic weight averaging for fluid-flow regression](https://mlsanity.com/paper/2109.08248/) (2021)
- [Physics-aware Reduced-order Modeling of Transonic Flow via β-Variational Autoencoder](https://mlsanity.com/paper/2205.00608/) (2022)
- [Residual-based physics-informed transfer learning: A hybrid method for accelerating long-term CFD simulations via deep learning](https://mlsanity.com/paper/2206.06817/) (2022)
- [Predicting the temporal dynamics of turbulent channels through deep learning](https://mlsanity.com/paper/2203.00974/) (2022)
- [A Review of Physics-based Machine Learning in Civil Engineering](https://mlsanity.com/paper/2110.04600/) (2021)
- [A Thermodynamics-informed Active Learning Approach to Perception and Reasoning about Fluids](https://mlsanity.com/paper/2203.05775/) (2022)
- [Emerging trends in machine learning for computational fluid dynamics](https://mlsanity.com/paper/2211.15145/) (2022)
- [Diff-SPORT: Diffusion-based Sensor Placement Optimization and Reconstruction of Turbulent flows in urban environments](https://mlsanity.com/paper/2506.00214/) (2025)
- [Generative Adversarial Reduced Order Modelling](https://mlsanity.com/paper/2305.15881/) (2023)
- [Diffusion model based data generation for partial differential equations](https://mlsanity.com/paper/2306.11075/) (2023)
- [Towards aerodynamic surrogate modeling based on β-variational autoencoders](https://mlsanity.com/paper/2408.04969/) (2024)
- [Data-Efficient Deep Operator Network for Unsteady Flow: A Multi-Fidelity Approach with Physics-Guided Subsampling](https://mlsanity.com/paper/2503.17941/) (2025)
- [Reduced-order modeling of unsteady fluid flow using neural network ensembles](https://mlsanity.com/paper/2402.05372/) (2024)
- [Enhancing Graph U-Nets for Mesh-Agnostic Spatio-Temporal Flow Prediction](https://mlsanity.com/paper/2406.03789/) (2024)
- [Machine Learning Applications to Computational Plasma Physics and Reduced-Order Plasma Modeling: A Perspective](https://mlsanity.com/paper/2409.02349/) (2024)
- [AeroJEPA: Learning Semantic Latent Representations for Scalable 3D Aerodynamic Field Modeling](https://mlsanity.com/paper/2605.05586/) (2026)
- [Data-driven Nonlinear Parametric Model Order Reduction Framework using Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2307.06816/) (2023)
- [Thermodynamics-informed super-resolution of scarce temporal dynamics data](https://mlsanity.com/paper/2402.17506/) (2024)
- [Decoder Decomposition for the Analysis of the Latent Space of Nonlinear Autoencoders With Wind-Tunnel Experimental Data](https://mlsanity.com/paper/2404.19660/) (2024)
- [EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations](https://mlsanity.com/paper/2504.07976/) (2025)
- [Predicting the wall-shear stress and wall pressure through convolutional neural networks](https://mlsanity.com/paper/2303.00706/) (2023)
- [Physics-Assisted Reduced-Order Modeling for Identifying Dominant Features of Transonic Buffet](https://mlsanity.com/paper/2305.13644/) (2023)

## Abstract (arXiv)

We propose a deep probabilistic-neural-network architecture for learning a minimal and near-orthogonal set of non-linear modes from high-fidelity turbulent-flow-field data useful for flow analysis, reduced-order modeling, and flow control. Our approach is based on β-variational autoencoders (β-VAEs) and convolutional neural networks (CNNs), which allow us to extract non-linear modes from multi-scale turbulent flows while encouraging the learning of independent latent variables and penalizing the size of the latent vector. Moreover, we introduce an algorithm for ordering VAE-based modes with respect to their contribution to the reconstruction. We apply this method for non-linear mode decomposition of the turbulent flow through a simplified urban environment, where the flow-field data is obtained based on well-resolved large-eddy simulations (LESs). We demonstrate that by constraining the shape of the latent space, it is possible to motivate the orthogonality and extract a set of parsimonious modes sufficient for high-quality reconstruction. Our results show the excellent performance of the method in the reconstruction against linear-theory-based decompositions. Moreover, we compare our method with available AE-based models. We show the ability of our approach in the extraction of near-orthogonal modes that may lead to interpretability.