# Deep Learning for Reduced Order Modelling and Efficient Temporal Evolution of Fluid Simulations

arXiv: https://arxiv.org/abs/2107.04556 · HTML: https://mlsanity.com/paper/2107.04556/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 46 references that are in the map.
- First posted on arXiv 9 Jul 2021.

## Builds on (most cited first, 5 of 5)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network](https://mlsanity.com/paper/1808.03314/) (2018)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014)
- [Prediction of wall-bounded turbulence from wall quantities using convolutional neural networks](https://mlsanity.com/paper/1912.12969/) (2019)

## Built on by (most cited first, 30 of 36)

- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations](https://mlsanity.com/paper/2211.00713/) (2022)
- [Physics Informed Token Transformer for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.08757/) (2023)
- [Physics-enhanced deep surrogates for partial differential equations](https://mlsanity.com/paper/2111.05841/) (2021)
- [A Finite Element-Inspired Hypergraph Neural Network: Application to Fluid Dynamics Simulations](https://mlsanity.com/paper/2212.14545/) (2022)
- [Multi-scale Time-stepping of Partial Differential Equations with Transformers](https://mlsanity.com/paper/2311.02225/) (2023)
- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [Predicting fluid-structure interaction with graph neural networks](https://mlsanity.com/paper/2210.04193/) (2022)
- [EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scale](https://mlsanity.com/paper/2510.24173/) (2025)
- [textitFastSVD-ML-ROM: A Reduced-Order Modeling Framework based on Machine Learning for Real-Time Applications](https://mlsanity.com/paper/2207.11842/) (2022)
- [Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations](https://mlsanity.com/paper/2402.17853/) (2024)
- [Adaptive learning of effective dynamics: Adaptive real-time, online modeling for complex systems](https://mlsanity.com/paper/2304.01732/) (2023)
- [Hyena Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2306.16524/) (2023)
- [PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs](https://mlsanity.com/paper/2405.08558/) (2024)
- [An Implicit GNN Solver for Poisson-like problems](https://mlsanity.com/paper/2302.10891/) (2023)
- [Geometry Matters: Benchmarking Scientific ML Approaches for Flow Prediction around Complex Geometries](https://mlsanity.com/paper/2501.01453/) (2024)
- [Deep Symmetric Autoencoders from the Eckart-Young-Schmidt Perspective](https://mlsanity.com/paper/2506.11641/) (2025)
- [Flowers: A Warp Drive for Neural PDE Solvers](https://mlsanity.com/paper/2603.04430/) (2026)
- [Digital Twin Data Modelling by Randomized Orthogonal Decomposition and Deep Learning](https://mlsanity.com/paper/2206.08659/) (2022)
- [MeshDQN: A Deep Reinforcement Learning Framework for Improving Meshes in Computational Fluid Dynamics](https://mlsanity.com/paper/2212.01428/) (2022)
- [Learning to simulate partially known spatio-temporal dynamics with trainable difference operators](https://mlsanity.com/paper/2307.14395/) (2023)
- [Fluid Viscosity Prediction Leveraging Computer Vision and Robot Interaction](https://mlsanity.com/paper/2308.02715/) (2023)
- [Inpainting Computational Fluid Dynamics with Deep Learning](https://mlsanity.com/paper/2402.17185/) (2024)
- [Principal Orthogonal Latent Components Analysis (POLCA Net)](https://mlsanity.com/paper/2410.07289/) (2024)
- [Operator learning for energy-efficient building ventilation control with computational fluid dynamics simulation of a real-world classroom](https://mlsanity.com/paper/2504.21243/) (2025)
- [Reduced-order modeling of Hamiltonian dynamics based on symplectic neural networks](https://mlsanity.com/paper/2508.11911/) (2025)
- [Elimination-compensation pruning for fully-connected neural networks](https://mlsanity.com/paper/2602.20467/) (2026)
- [Learning Permutation-invariant Macroscopic Dynamics](https://mlsanity.com/paper/2605.30812/) (2026)

## Abstract (arXiv)

Reduced Order Modelling (ROM) has been widely used to create lower order, computationally inexpensive representations of higher-order dynamical systems. Using these representations, ROMs can efficiently model flow fields while using significantly lesser parameters. Conventional ROMs accomplish this by linearly projecting higher-order manifolds to lower-dimensional space using dimensionality reduction techniques such as Proper Orthogonal Decomposition (POD). In this work, we develop a novel deep learning framework DL-ROM (Deep Learning - Reduced Order Modelling) to create a neural network capable of non-linear projections to reduced order states. We then use the learned reduced state to efficiently predict future time steps of the simulation using 3D Autoencoder and 3D U-Net based architectures. Our model DL-ROM is able to create highly accurate reconstructions from the learned ROM and is thus able to efficiently predict future time steps by temporally traversing in the learned reduced state. All of this is achieved without ground truth supervision or needing to iteratively solve the expensive Navier-Stokes(NS) equations thereby resulting in massive computational savings. To test the effectiveness and performance of our approach, we evaluate our implementation on five different Computational Fluid Dynamics (CFD) datasets using reconstruction performance and computational runtime metrics. DL-ROM can reduce the computational runtimes of iterative solvers by nearly two orders of magnitude while maintaining an acceptable error threshold.