A Physics-Informed Spatiotemporal Deep Learning Framework for Turbulent Systems

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-Benard 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.

💡 Research Summary

This paper introduces a physics‑informed spatiotemporal deep learning surrogate for Rayleigh‑Bénard convection (RBC), a prototypical turbulent thermal flow. The authors combine a convolutional autoencoder (CAE) for spatial dimensionality reduction with a ConvLSTM‑based sequence‑to‑sequence architecture that draws inspiration from large language models to capture long‑range temporal dependencies. Training is guided by a physics‑informed loss that penalizes violations of the Navier‑Stokes equations under the Boussinesq approximation, thereby enforcing mass, momentum, and energy conservation. To address the stochastic nature of turbulent dynamics, the model incorporates conformal prediction, providing distribution‑free, calibrated prediction intervals for each spatial point and time step.

The dataset consists of two‑dimensional DNS of RBC at a high Rayleigh number (Ra = 2.54 × 10⁸) on a 256 × 256 grid, with four fields (horizontal velocity, vertical velocity, pressure, temperature) sampled every 0.1 s for 2 500 time steps. The CAE compresses the full‑resolution state Yₜ ∈ ℝ^{256×256×4} into a latent representation ˜Yₜ ∈ ℝ^{32×32×16} through five strided convolutional layers followed by a symmetric decoder. The ConvLSTM receives a sequence of b = 10 latent frames and predicts a sequence of a = 20 future frames, using a recurrent gating mechanism that retains spatial locality while modeling temporal evolution.

The loss function combines three terms: (1) an L₂ reconstruction loss between decoded outputs and DNS data, (2) a PDE residual loss computed via automatic differentiation of the Navier‑Stokes and energy equations, and (3) an L₂ regularization on network weights. Hyper‑parameters λ₁ = 0.1 (PDE weight) and λ₂ = 1e‑4 (regularization) balance physical fidelity against data fit.

Uncertainty quantification is achieved through conformal prediction. Residuals from a held‑out validation set are used to construct non‑conformity scores; the (1 − α) quantile of these scores defines a symmetric prediction band around each forecast. With α = 0.1, empirical coverage exceeds 92 %, surpassing the nominal 90 % level.

Evaluation focuses on two aspects. First, statistical fidelity: the surrogate reproduces key diagnostics of turbulent convection, including kinetic energy spectra, temperature‑velocity correlation functions, and Nusselt number trends, even for forecasts extending beyond 100 time steps. Second, computational efficiency: on a single RTX 3090 GPU the model generates roughly 5 000 frames per second, yielding a speed‑up of 20–50× compared with the original DNS run on a high‑performance cluster. Ablation studies show that removing the physics‑informed term degrades RMS error by 45 % and leads to drift in conserved quantities.

The authors acknowledge limitations: the work is confined to two‑dimensional flows; extending to three dimensions will increase memory and compute demands dramatically. The PDE residual term assumes a fixed grid and boundary conditions, which may reduce flexibility for irregular domains. Conformal intervals, while valid, can be overly conservative, potentially masking finer‑grained uncertainty. Future directions include integrating multi‑scale operator learning (e.g., Koopman or Fourier neural operators) with the current framework, learning adaptive weighting of physics penalties, and deploying the surrogate in online control or data‑assimilation pipelines.

In summary, the proposed Physics‑Informed Convolutional Recurrent Neural Network (PI‑CRNN) delivers a statistically rigorous, physically consistent, and computationally cheap surrogate for high‑Rayleigh‑number turbulent convection, opening pathways for large‑scale climate, engineering, and scientific simulations where full DNS is infeasible.