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Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow

Steffen Wiewel, Moritz Becher, Nils Thuerey

arXiv:1802.10123v3cs.LGcs.GR

TL;DR

The paper addresses whether neural networks can predict temporal evolution of dense three-dimensional fluid pressure fields despite the high dimensionality of Eulerian space-time data. It combines learned autoencoder latent spaces with an LSTM-based prediction network, achieving robust fluid simulations and a reported 155× speedup over a parallelized iterative pressure solver for a 128^3 volume. The authors identify autoencoder quality and limited small-scale dynamics as important areas for improvement.

  • Problem

    The paper asks whether neural networks can predict temporal evolution of complex three-dimensional fluid pressure fields, a question not previously addressed in the cited work.

  • Method

    The method compresses three-dimensional fluid fields into autoencoder latent spaces and uses an LSTM-based recurrent network to predict their temporal evolution.

  • Results

    155× speedup was measured for pressure inference plus encoding and decoding on a 128^3 volume versus a parallelized MIC-CG pressure solver, while the solver retained higher overall accuracy.

  • Takeaways & Limitations

    Learned latent spaces with LSTM-CNN hybrids can predict dense physical-function evolution and substantially increase simulation performance across fluid scenes.

  • Takeaways & Limitations

    The current LSTM relies strongly on an autoencoder that primarily captures large-scale dynamics, while the autoencoder can introduce noise and small-scale dynamics remain limited.

Abstract

from arXiv · show

We propose a method for the data-driven inference of temporal evolutions of physical functions with deep learning. More specifically, we target fluid flows, i.e. Navier-Stokes problems, and we propose a novel LSTM-based approach to predict the changes of pressure fields over time. The central challenge in this context is the high dimensionality of Eulerian space-time data sets. We demonstrate for the first time that dense 3D+time functions of physics system can be predicted within the latent spaces of neural networks, and we arrive at a neural-network based simulation algorithm with significant practical speed-ups. We highlight the capabilities of our method with a series of complex liquid simulations, and with a set of single-phase buoyancy simulations. With a set of trained networks, our method is more than two orders of magnitudes faster than a traditional pressure solver. Additionally, we present and discuss a series of detailed evaluations for the different components of our algorithm.

1. Introduction

The paper reframes temporal prediction of complex fluid functions as a deep-learning problem, targeting pressure-field evolution in three dimensions. It introduces latent-space prediction as a route to faster simulations and evaluates an LSTM-based architecture for dense physical functions.

  • Motivation: Traditional transient simulations require finely resolving high-dimensional physical functions across space and time.This creates many degrees of freedom in the numerical representation.
  • Motivation: The paper targets temporal evolution of complex fluid-flow functions, especially three-dimensional pressure fields.Fluids are presented as important physical systems for learning-based models.
  • Research gap: Few prior works addressed temporal evolution of physical systems, and prediction of complex three-dimensional fluid pressure fields had not previously been addressed.Earlier machine-learning studies considered local interactions, single-time projections, high-resolution flows, Lagrangian objects, or two-dimensional control.
  • Approach: The method reduces the original three-dimensional problem to a smaller spatial latent space and learns mappings between the original and reduced representations.A second network predicts from collections of reduced representations.
  • Contributions: The paper contributes an LSTM architecture for dense physical 3D functions, an efficient strongly compressive encoder-decoder, and evaluations of training modalities.These components form the basis of a fast neural-network simulation algorithm.

2. Related Work and Background

The background situates the work within neural-network function approximation, latent representations, and machine learning for physics. It focuses on incompressible Navier-Stokes flow and Eulerian pressure solving, where pressure computation is often especially expensive.

  • Neural networks: Neural networks approximate unknown functions by optimizing a loss over learnable representation parameters.The paper describes a network layer using activations, weight matrices, and biases.
  • Latent representations: Latent spaces provide nonlinear representations tied to data distributions, and the paper uses autoencoder latent spaces to predict dense physical-function evolution.The discussion connects this approach to prior latent-space methods in image processing and synthesis.
  • Physics learning: Prior physics-learning work included predictions of two-dimensional physics, rigid-body dynamics, liquid motion for robotic control, and adversarial inference of solutions.These studies largely concern lower-dimensional or different physical prediction settings.
  • Fluid model: The paper uses the incompressible Navier-Stokes model for flow physics, with velocity and pressure as its most important quantities.Density, kinematic viscosity, and external forces are also identified as model parameters.
  • Fluid model: For liquids, a signed-distance function is advected with the flow or reconstructed from advected particles.This supplies the liquid-surface representation used in the simulation setting.
  • Numerical setting: The paper focuses on Eulerian grid-based solvers because CNNs are particularly amenable to grid discretizations.Pressure receives special attention because its iterative solver is often the most expensive algorithmic component.

3. Method

The method predicts dense Eulerian physical fields by first compressing spatial data into a learned latent space, then modeling temporal evolution with an LSTM-based prediction network. A hybrid recurrent-convolutional design supports prediction of future latent states while reducing computational and parameter costs.

  • Problem: Dense Eulerian functions such as pressure fields or velocity fields can contain millions of spatial degrees of freedom when discretized.The framework represents physical functions over three-dimensional space and time, with scalar pressure and vectorial velocity as examples.
  • Spatial dimensionality reduction: A convolutional autoencoder maps each three-dimensional input x(t) to a lower-dimensional latent encoding c_t and reconstructs it through a decoder.The encoder and decoder are trained to reconstruct x accurately, and the latent dimensionality is selected to make dense three-dimensional prediction feasible.
  • Spatial dimensionality reduction: The autoencoder uses six convolutional layers in both encoder and decoder, a bottleneck of size m_s, and achieves a total reduction factor of 256.The architecture uses LeakyReLU activations and increases or decreases feature counts by factors of two across layers.
  • Autoencoder training: Greedy layer-wise pretraining adds encoder-decoder depth progressively, with only the deepest path remaining active after pretraining.For typical depth l = 5, the authors perform six pretraining runs before training the complete model.
  • Temporal prediction: Because latent-vector entries have no meaningful spatial ordering, the predictor uses fully connected LSTM units together with convolutions that translate the LSTM state into latent-space outputs.The hybrid structure is proposed to limit the rapid growth of LSTM weights and overfitting risk associated with fully recurrent designs.
  • Temporal prediction: The temporal prediction network recurrently encodes n+1 latent states into a temporal context d and decodes that context into o future latent states.Training minimizes the mean absolute error between predicted and ground-truth future latent states using an L1 norm.
  • Scope and training constraints: Changing the latent-space size m_s requires retraining the prediction network because its input and internal context dimensions are fixed.The autoencoder can accept varying input sizes, but the prediction network is trained for fixed latent-space inputs and context sizes.

4. Fluid Flow Data

The paper generates fluid-flow data with a Navier–Stokes solver and evaluates velocity, total pressure, and split-pressure inference across complex liquid and smoke simulations. The data are strongly time-varying and include randomized, augmented three-dimensional scenes.

  • Training data are generated with operator splitting, including advection, external-force evaluation, and pressure computation.
  • The split-pressure formulation decomposes total pressure into hydrostatic and dynamic components, pt = ps + pd.The hydrostatic component can be evaluated efficiently, while the dynamic component captures residual fluctuations.
  • The framework targets velocity u, total pressure pt, and split pressure (ps, pd), with pressure inference replacing the pressure solve while retaining advection and velocity correction.The pressure solve is described as typically the most time-consuming part of the simulation.
  • The evaluation uses randomized scenes with strong splashes, vortices, and typically large CFL numbers around 2-3.
  • The data sets include liquid64, liquid128, and smoke128, with liquid scenes involving sloshing waves and collisions and smoke scenes containing buoyant plumes.Rotational mirroring augmentation produces data-set sizes from 80k to 400k entries.

5. Evaluation and Training

The evaluation measures reconstruction and temporal prediction quality using PSNR and surface error across liquid simulations. Results show accurate pressure prediction, effective interval prediction, efficient multi-step output, and a smaller hybrid architecture than the fully recurrent alternative.

  • Evaluation combines PSNR with a surface-based Hausdorff distance to assess both field quality and liquid-interface position.
  • 69.14 average PSNR is achieved by total-pressure encoding, while velocity encoding produces significantly larger reconstruction errors.
  • 64.80 average PSNR results from autoencoder-plus-prediction inference, only 6.2% below the autoencoder baseline.The LSTM does not benefit from the VAE’s normalized latent space in these tests.
  • After 100 steps, interval prediction with ip = 14 keeps average pressure surface error below two cells.The interval scheme alternates network prediction with a full simulation step.
  • Three-step prediction requires only 30% more evaluation time than one-step prediction despite producing three times as many outputs.Accuracy barely degrades when predicting multiple steps from one input context.
  • The hybrid LSTM outperforms the fully recurrent version while using 8.9m fewer weights; for 128^3 data, the hybrid model uses 64m weights versus an infeasible fully recurrent design.The reported comparison gives 19.5m versus 28.4m weights for one tested hybrid and fully recurrent pair.

6. Results

The method predicts pressure-field evolution across higher-resolution liquid and smoke simulations while providing substantial speedups over iterative pressure solvers. Its accuracy remains lower than regular solvers and depends on autoencoder quality and prediction settings.

  • Higher-resolution liquid results: 44.8 average PSNR was achieved for liquid128 pressure-field predictions despite eight times more degrees of freedom per volume.The model also handled an unseen configuration and a 650-step scene that successfully came to rest.
  • Higher-resolution smoke results: Smoke128 predictions captured vortex evolution but underestimated pressure values and reduced small-scale motions.The authors identify clear room for improvement for this data set.
  • Performance: 155× speedup was measured for 128^3 pressure inference, including encoding and decoding, versus an eight-thread parallelized MIC-CG solver.Prediction, encoding, and decoding took 9.5ms, 4.1ms, and 3.3ms, respectively.
  • Accuracy-speed trade-off: The learned method trades accuracy for speed, with degradation relative to a regular iterative solver controlled by the prediction interval.A simple implementation still achieved 10x speedup for interval prediction with ip = 14.
  • Limitations and future improvements: Improving the autoencoder is important because its representation and noise affect the quality of temporal predictions.The authors also report that larger data sets should directly improve predictions, especially where latent-space data are difficult to augment.

7. Conclusions

The paper concludes that learned latent spaces combined with LSTM-CNN hybrids can predict dense physical functions over time. This yields a data-driven solver with practical speedups, while remaining an initial step requiring further accuracy and performance work.

  • Conclusions: Deep neural networks can successfully predict the temporal evolution of dense physical functions.
  • Conclusions: A long-running liquid128 prediction successfully came to rest after 800 time steps.
  • Conclusions: Learned latent spaces combined with LSTM-CNN hybrids are suitable for predicting dense physical functions.
  • Conclusions: More than 150x faster than a regular pressure solve was achieved at the core of the data-driven solver.The conclusion describes this as a practical speedup for a neural-network-based simulation algorithm.

Appendix B: Additional Results

Additional results compare inferred quantities and show temporal predictions across liquid and smoke scenes. Pressure-based models capture large-scale liquid motion more closely than the velocity version, while divergence errors remain small relative to the reference solver.

  • Additional scenes: Additional renderings show liquid64, liquid128, and smoke128 models at different points in time.
  • Additional comparisons: Pressure versions closely captured large-scale liquid motions over 40 time steps, whereas the velocity version produced large surface-position errors.Split-pressure and especially VAE models introduced artifacts at smaller scales.
  • Divergence behavior: 2.1 · 10^-4 divergence increase per time step was introduced on average by the LSTM pressure field, versus 3.1 · 10^-3 residual divergence for the numerical solver.The authors report no significant mass-conservation changes compared with reference simulations when using interval predictions.
  • Visualization: A reduced two-dimensional autoencoder visualization compares ground truth, an autoencoder baseline, and temporal prediction by the LSTM network.

FLIP Simulation

The liquid data sets are generated with a FLIP-based Navier-Stokes setup using fixed time steps, gravity, and conjugate-gradient pressure solves. FLIP also forms the base simulation for the neural-network-driven interval prediction.

  • Simulation setup: The liquid datasets use a classic Navier-Stokes solver with timestep 0.1 and conjugate-gradient pressure accuracy of 5 · 10^-5.
  • Simulation setup: The setup applies only gravity, using vector (0.0,−0.01,0.0), with no additional viscosity or surface-tension forces.
  • Additional examples: Several smoke128 scenes are predicted with the LSTM network using an interval of ip = 3.
  • FLIP Simulation: FLIP represents a grid-particle hybrid used both to generate liquid datasets and to support the neural-network-driven interval prediction simulation.
  • FLIP Simulation: Each simulation step advects particles, the level set, and velocity, then constructs and merges a particle-surface level set before narrow-band extrapolation.

Prediction Integration

The framework predicts several simulation fields from learned representations, allowing selected numerical simulation steps to be replaced by neural-network inference. Examples include long-horizon pressure predictions across liquid datasets and complex initial configurations.

  • Supported prediction fields: The framework supports predicting velocity, total pressure, and decomposed hydrostatic and dynamic pressure fields from FLIP simulations.The compared architectures target different fields, including u, p, ps, and pd.
  • Prediction examples: 30 prediction steps at resolution 642 show that LSTM predictions closely follow the temporal evolution of ground-truth pressure fields.The comparison includes ground truth, an autoencoder baseline, and LSTM predictions over a long horizon with ip = ∞.
  • Prediction examples: The evaluation includes liquid64 and liquid128 initial states, with more complex initial shapes visible in several liquid128 configurations.These figures document the range of initial scene states used for liquid simulations.
  • Integration into simulation: For pressure prediction, neural inference replaces the pressure solve while advection and pressure-gradient-based velocity correction remain necessary.Predicting velocity can omit additional steps, including velocity advection.

Appendix D: Hyperparameters

The appendix searches prediction-network hyperparameters on a subset of training scenes and evaluates the selected configuration on independently generated test scenes. The reported search identifies favorable settings and a low test-set prediction error.

  • Search procedure: Hyperparameter searches vary prediction-network parameters across training runs on a subset of 100 training scenes.Each run is evaluated using training and validation errors.
  • Search procedure: Search plots encode mean absolute training error by color and validation error by circle size, with the two best runs highlighted.Purple denotes the best training error and yellow the worst.
  • Evaluation context: The appendix uses training-history and example-data figures to document optimization behavior and the tested simulation data.The smoke128 examples show states at t = 65, while Fig. 21 shows training history for the liquid128 pt network.
  • Search findings: The network performed best without weight-decay regularization in one reported hyperparameter search.This observation is specifically reported for Fig. 20a.
  • Generalization evaluation: 0.0201 mean absolute prediction error was obtained on an independently generated test set of 40 scenes.The result is reported for the training-history example associated with Fig. 21.
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