# An artificial neural network approach to bifurcating phenomena in computational fluid dynamics

arXiv: https://arxiv.org/abs/2109.10765 · HTML: https://mlsanity.com/paper/2109.10765/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 64 references that are in the map.
- First posted on arXiv 22 Sep 2021.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equations](https://mlsanity.com/paper/2106.05102/) (2021)

## Built on by (most cited first, 11 of 11)

- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applications](https://mlsanity.com/paper/2406.03569/) (2024)
- [A practical existence theorem for reduced order models based on convolutional autoencoders](https://mlsanity.com/paper/2402.00435/) (2024)
- [Non-linear manifold ROM with Convolutional Autoencoders and Reduced Over-Collocation method](https://mlsanity.com/paper/2203.00360/) (2022)
- [Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations](https://mlsanity.com/paper/2407.19707/) (2024)
- [Neural empirical interpolation method for nonlinear model reduction](https://mlsanity.com/paper/2406.03562/) (2024)
- [Optimal Transport-inspired Deep Learning Framework for Slow-Decaying Kolmogorov n-width Problems: Exploiting Sinkhorn Loss and Wasserstein Kernel](https://mlsanity.com/paper/2308.13840/) (2023)
- [Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decomposition](https://mlsanity.com/paper/2511.23037/) (2025)
- [Operator-based machine learning framework for generalizable prediction of unsteady treatment dynamics in stormwater infrastructure](https://mlsanity.com/paper/2507.04682/) (2025)
- [Adaptive Sampling for Hydrodynamic Stability](https://mlsanity.com/paper/2512.13532/) (2025)
- [Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs](https://mlsanity.com/paper/2601.11259/) (2026)

## Abstract (arXiv)

This work deals with the investigation of bifurcating fluid phenomena using a reduced order modelling setting aided by artificial neural networks. We discuss the POD-NN approach dealing with non-smooth solutions set of nonlinear parametrized PDEs. Thus, we study the Navier-Stokes equations describing: (i) the Coanda effect in a channel, and (ii) the lid driven triangular cavity flow, in a physical/geometrical multi-parametrized setting, considering the effects of the domain's configuration on the position of the bifurcation points. Finally, we propose a reduced manifold-based bifurcation diagram for a non-intrusive recovery of the critical points evolution. Exploiting such detection tool, we are able to efficiently obtain information about the pattern flow behaviour, from symmetry breaking profiles to attaching/spreading vortices, even at high Reynolds numbers.