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Deep Learning for Reduced Order Modelling and Efficient Temporal Evolution of Fluid Simulations
Pranshu Pant, Ruchit Doshi, Pranav Bahl, Amir Barati Farimani
TL;DR
High-fidelity simulations are costly in real-time and multi-query settings, and conventional projection-based ROMs rely predominantly on linear subspaces. DL-ROM learns nonlinear reduced states and uses them to reconstruct flow fields and predict future steps; the paper reports accurate reconstructions and computationally efficient prediction.
Problem
High-dimensional full-order simulations impose substantial computational burdens in real-time or multi-query settings, while projection-based ROMs are predominantly linear.
Source: §1 · pp. 2–3Method
DL-ROM learns nonlinear reduced-state embeddings and uses 3D convolutional architectures to reconstruct flow states and predict future steps.
Source: §1 · p. 3, §3.2 · p. 9, §3.2.1 · p. 10Results
The learned reduced state supports highly accurate reconstructions and computationally efficient prediction of future flow states.
Source: §5 · p. 13Takeaways & Limitations
The approach predicts future time steps from learned reduced states without solving the computationally expensive Navier–Stokes equations.
Source: §1 · p. 3
Abstract
from arXiv · showhide
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.
1 Introduction:
High-fidelity simulations create computational burdens in real-time and multi-query settings, while projection-based ROMs reduce dimensions through predominantly linear subspaces. This work proposes nonlinear neural reduced states for reconstructing and predicting future CFD flow fields.
1 Introduction:
- The evaluated datasets cover different simulation conditions and contain two-dimensional u-velocity data in the X direction.The figure lists sea-surface temperature, bluff-body flows, channel flow, and a custom OpenFOAM dataset.Source: Figure 1 · p. 2
- High-dimensional full-order simulations can be prohibitively costly in real-time or multi-query settings, motivating reduced representations that preserve physical features with minimal information loss.Projection-based ROMs form low-dimensional trial subspaces from snapshots, but are predominantly linear.Source: §1 · p. 2
- POD produces a linear ROM from snapshot-matrix decomposition, while DMD represents high-order data in a low-dimensional subspace with defined dynamics.Other approaches discussed include POD-GP, Petrov-Galerkin, and Koopman operator theory.Source: §1 · pp. 2–3
- DL-ROM uses a neural network for nonlinear dimensionality reduction, learning reduced embeddings that support high-fidelity reconstruction and future-step prediction without solving Navier–Stokes equations.Its model takes concatenated state vectors from previous time steps as input and predicts the state at t+1.Source: §1 · p. 3
2 Datasets
The study benchmarks its approach on five datasets with different simulation conditions. The cases span three two-dimensional flows, DNS channel flow, and global sea-surface temperature data.
2.1.1 Flow around 2D Circular Cylinder
- The circular-cylinder case models Re = 100 flow and uses 2,082 snapshots of the periodic vortex region for training.The simulation uses a RANS approach and recovers the Z-vorticity field.Source: §2.1.1 · p. 4
2.1.2 Flow around 2D Square Cylinder
- The square-cylinder case models Re = 100 flow and recovers velocity from the temporal information in previous 9 snapshots.The training set contains 4,087 snapshots.Source: §2.1.2 · pp. 4–5
2.1.3 Flow over 2D Plate
- The 2D plate case uses RANS with a k-epsilon turbulence model and targets vorticity magnitude from temporal snapshots.The training data comprises 1,500 snapshots.Source: §2.1.3 · p. 5
2.2 2D Channel Flow dataset
- The channel-flow dataset is DNS data sampled on a uniform 512 x 128 grid at 2,500 timesteps.The source data comes from the Johns Hopkins Turbulence Database.Source: §2.2 · p. 5
2.3 NOAA Optimum Interpolation (OI) SST, V2
- The NOAA OI SST dataset uses weekly global data from 1981–2011, with 2,000 snapshots on a 1-degree latitude-by-longitude grid.The data are sea-surface temperatures in degrees Celsius.Source: §2.3 · p. 6
3.1 Existing Approach
The existing approaches combine autoencoders for reduced states with sequence models for future-step prediction. The section outlines autoencoder variants and describes LSTMs as a means of retaining temporal information.
3.1 Existing Approach
- Existing methods use autoencoders to reconstruct the current flow state and convolutional LSTM autoencoders to predict future states.The prior pipeline combines three separate models to produce reduced states and predict future timesteps.Source: §3.1 · p. 6
3.1.1 Autoencoders
- Autoencoders compress a current CFD state into reduced states, with nonlinear mappings offering an alternative to POD’s linear projection.Figure 5 depicts iterative prediction by feeding each predicted t+1 result back into the 10-timestep input.Source: §3.1.1 · pp. 6–7, Figure 5 · p. 7
3.1.2 Multi-Layered Perceptron Autoencoder:
- A Multi-Layered Perceptron Autoencoder encodes a one-dimensional flow vector into a bottleneck representing compressed reduced states, then decodes it to the original space.Both encoder and decoder use fully connected layers.Source: §3.1.2 · p. 7
3.1.3 Convolutional Autoencoders:
- Convolutional Autoencoders replace fully connected layers with convolutions, then use transpose convolutions to reconstruct the higher-dimensional input.The encoder’s extracted features pass through a fully connected bottleneck layer.Source: §3.1.3 · p. 7
3.1.4 Long Short-Term Memory networks
- LSTMs use input, forget, and output gates plus a memory cell to regulate information flow and retain information over longer sequences.The section presents them as addressing the vanishing-gradient and memory-retention drawbacks of vanilla RNNs.Source: §3.1.4 · pp. 7–8
3.2 Our Approach
DL-ROM uses 3D convolutions and learned reduced states to predict future fluid states, avoiding the spatial-information loss associated with vectorized LSTM inputs. A 3D convolutional autoencoder and U-Net support this approach, with predictions also evaluated in an unsupervised 20-step rollout.
3.2 Our Approach
- On five datasets, Figure 7 evaluates validation predictions from an initial-timestep input as negative-log MSE traces; declining traces indicate increasing MSE over time.Predictions are recursively generated without ground-truth supervision, so errors accumulate over the rollout.Source: Figure 7 · p. 9
- DL-ROM replaces vectorized LSTM temporal processing, which loses spatial correlations, with 3D convolutions that extract spatial and temporal features.The approach uses 3D convolutions instead of LSTMs, which are described as difficult and computationally expensive to train and prone to overfitting and exploding gradients.Source: §3.2 · pp. 8–9
3.2.1 3D Convolutional Autoencoders
- The 3D convolutional autoencoder compresses concatenated prior frames into a nonlinear reduced state, then decodes it into the predicted next flow state.Its encoder and decoder each use five 3D convolutional layers; depth-wise separable convolutions reduce learned parameters while maintaining similar feature-extraction performance.Source: §3.2.1 · p. 10
3.2.2 3D U-Net
- The U-Net links encoder and decoder layers at matching resolutions, combining spatial features to localize regions of interest more precisely.The model stacks 10 frames sampled every 10 timesteps to predict the 100th timestep, and trains against that target using MAE.Source: §3.2.2 · p. 10
- Figure 8 compares training time per epoch and MAE with and without transfer learning from the 2D cylinder model, reporting faster training with comparable performance on most datasets.The transfer-learning setup trains only the initial and final DL-ROM layers.Source: Figure 8 · p. 11
- The objective minimizes MAE for the next state given the preceding 10 frames, and a 20-step rollout feeds each prediction back into subsequent inputs.The rollout is evaluated against ground truth for accuracy and runtime, without ground-truth supervision during prediction.Source: §3.2.2 · p. 11
4 Results
Across the evaluated datasets, DL-ROM predicts future flow states with low error and substantially shorter iteration runtimes than CFD solvers. Results vary by flow structure, and transfer learning speeds training while generally retaining comparable performance.
4 Results
- Models are trained and validated using an approximately 6:1 split, with MAE loss and most network layers held constant across datasets.Dataset-specific input sizes are handled in the encoder’s initial and decoder’s final layers.Source: §4 · p. 11
- Transfer learning from the 2D cylinder model speeds training while maintaining comparable performance to conventional training on most datasets.Only the initial and final network layers are trained from scratch for each dataset.Source: §4 · p. 12
- A latent size of 32 gives the best reconstruction performance among the tested sizes while reducing computational overhead for the reduced-state representation.The prediction input uses 10 previous frames, each separated by 10 timesteps; the frame spacing is a variable hyperparameter.Source: §4 · p. 12
- Nearly 2 orders of magnitude faster than CFD simulations across three datasets, DL-ROM also maintains low prediction error, particularly on 2D cylinder, airfoil, and plate flows.The runtime comparison uses OpenFOAM; the reported near-two-orders reduction applies to the three custom datasets used for runtime comparison.Source: §4 · p. 12, Figure 9 · p. 12
- DL-ROM accurately predicts von-Karman vortex streets, but high-gradient obstruction regions are difficult and small channel-flow structures are lost during compression.The reported negative-log MSE values are evaluated per pixel on validation data.Source: §4 · p. 12
5 Conclusions
DL-ROM learns reduced-order embeddings and uses them to predict future fluid states with 3D autoencoder and U-Net architectures. The authors report accurate reconstructions, nearly two orders of magnitude lower runtimes than OpenFOAM solvers, and low MSE over 20 unsupervised rollout iterations.
5 Conclusions
- DL-ROM combines 3D autoencoders and U-Nets to encode flow snapshots from prior timesteps into reduced states and predict future full-order flow states.The method also uses depth-wise separable convolutions and transfer learning.Source: §5 · p. 13
- The authors report accurate future-flow predictions across five datasets, with minor differences concentrated mostly around high-gradient regions.The conclusion also reports low MSE through 20 iterations of prediction without ground-truth supervision.Source: §5 · p. 13
- DL-ROM reduces computational runtimes by nearly two orders of magnitude compared with OpenFOAM CFD solvers.The authors propose handing later iterations to DL-ROM after initial CFD iterations, or interleaving DL-ROM predictions with sparse CFD evaluations.Source: §5 · p. 13