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A Physics-Informed Spatiotemporal Deep Learning Framework for Turbulent Systems

Luca Menicali, Andrew Grace, David H. Richter, Stefano Castruccio

TL;DR

This work tackles long-range forecasting of turbulent RBC with prohibitive DNS cost by introducing PI-CRNN, a physics-informed spatiotemporal surrogate. The model combines a convolutional autoencoder for spatial compression, a sequence-to-sequence ConvLSTM for reduced-space dynamics, PDE-based penalties for physical fidelity, and conformal prediction for distribution-free uncertainty quantification. It achieves superior spatial reconstruction compared to linear methods and yields physically consistent, long-range forecasts with quantified uncertainty, while offering substantial computational savings (e.g., 7 turnover forecasts in under 20 s vs DNS). The framework is extensible to other nonlinear thermofluid systems and can be augmented with advanced architectures like vision transformers to capture global spatiotemporal interactions.

Abstract

Fluid thermodynamics underpins atmospheric dynamics, climate science, industrial applications, and energy systems. However, direct numerical simulations (DNS) of such systems can be computationally prohibitive. To address this, we present a novel physics-informed spatiotemporal surrogate model for Rayleigh-Bénard convection (RBC), a canonical example of convective fluid flow. Our approach combines convolutional neural networks, for spatial dimension reduction, with an innovative recurrent architecture, inspired by large language models, to model long-range temporal dynamics. Inference is penalized with respect to the governing partial differential equations to ensure physical interpretability. Since RBC exhibits turbulent behavior, we quantify uncertainty using a conformal prediction framework. This model replicates key physical features of RBC dynamics while significantly reducing computational cost, offering a scalable alternative to DNS for long-term simulations.

A Physics-Informed Spatiotemporal Deep Learning Framework for Turbulent Systems

TL;DR

This work tackles long-range forecasting of turbulent RBC with prohibitive DNS cost by introducing PI-CRNN, a physics-informed spatiotemporal surrogate. The model combines a convolutional autoencoder for spatial compression, a sequence-to-sequence ConvLSTM for reduced-space dynamics, PDE-based penalties for physical fidelity, and conformal prediction for distribution-free uncertainty quantification. It achieves superior spatial reconstruction compared to linear methods and yields physically consistent, long-range forecasts with quantified uncertainty, while offering substantial computational savings (e.g., 7 turnover forecasts in under 20 s vs DNS). The framework is extensible to other nonlinear thermofluid systems and can be augmented with advanced architectures like vision transformers to capture global spatiotemporal interactions.

Abstract

Fluid thermodynamics underpins atmospheric dynamics, climate science, industrial applications, and energy systems. However, direct numerical simulations (DNS) of such systems can be computationally prohibitive. To address this, we present a novel physics-informed spatiotemporal surrogate model for Rayleigh-Bénard convection (RBC), a canonical example of convective fluid flow. Our approach combines convolutional neural networks, for spatial dimension reduction, with an innovative recurrent architecture, inspired by large language models, to model long-range temporal dynamics. Inference is penalized with respect to the governing partial differential equations to ensure physical interpretability. Since RBC exhibits turbulent behavior, we quantify uncertainty using a conformal prediction framework. This model replicates key physical features of RBC dynamics while significantly reducing computational cost, offering a scalable alternative to DNS for long-term simulations.
Paper Structure (17 sections, 18 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 17 sections, 18 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction & Background
  2. Data
  3. Methods
  4. Spatial Model: Convolutional Autoencoder
  5. Temporal Model: Recurrent Neural Network
  6. Spatiotemporal Model: Convolutional Recurrent Neural Network
  7. Inference
  8. Training and Testing set
  9. Hyperparameters
  10. Convolutional Autoencoder Inference
  11. Recurrent Neural Network Physics-Informed Inference
  12. Uncertainty Quantification
  13. Results
  14. Spatial Analysis (Dimension Reduction)
  15. Temporal Analysis
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Architecture of the CAE model. The input $\bm{Y}$ is projected onto reduced space through a series of strided convolutions, \ref{['eq:convolution']} in yellow, to $\bm{\tilde{Y}} \in \mathbb{R}^{\Tilde{n}_{x} \times \Tilde{n}_{z} \times \tilde{d}}$ and subsequently projected back onto the original dimensions, $\mathbb{R}^{n_{x} \times n_{z} \times d}$, through a series of strided deconvolutions, in blue.
  • Figure 2: CAE results for one observation in the validation set, $t=2{,}125$, showing the data (top row) and the reconstruction (bottom row) after reducing the dimensions of the data by 93.75%. The $\text{MSE}_{\text{space}}$ and SSIM for this observation are $6.12 \times 10^{-6}$ and $0.99$, respectively.
  • Figure 3: Snapshots of the temperature forecast over the first turnover time ($\approx$ 5.5s), where the spatiotemporal dynamics remain strongly dependent on the input sequence. An animation is available in the online code associated with the manuscript.