[Paper Review] Physics-aware deep neural networks for surrogate modeling of turbulent natural convection

Didier Lucor, Atul Agrawal+1 more|arXiv (Cornell University)|Mar 5, 2021

Model Reduction and Neural Networks35 references29 citations

TL;DR

The paper investigates physics-informed neural networks (PINNs) as surrogates for turbulent Rayleigh-Bénard convection, introducing padding and relaxed incompressibility to improve training; it reports high accuracy using limited DNS data even at very high Rayleigh numbers.

ABSTRACT

Recent works have explored the potential of machine learning as data-driven turbulence closures for RANS and LES techniques. Beyond these advances, the high expressivity and agility of physics-informed neural networks (PINNs) make them promising candidates for full fluid flow PDE modeling. An important question is whether this new paradigm, exempt from the traditional notion of discretization of the underlying operators very much connected to the flow scales resolution, is capable of sustaining high levels of turbulence characterized by multi-scale features? We investigate the use of PINNs surrogate modeling for turbulent Rayleigh-B{é}nard (RB) convection flows in rough and smooth rectangular cavities, mainly relying on DNS temperature data from the fluid bulk. We carefully quantify the computational requirements under which the formulation is capable of accurately recovering the flow hidden quantities. We then propose a new padding technique to distribute some of the scattered coordinates-at which PDE residuals are minimized-around the region of labeled data acquisition. We show how it comes to play as a regularization close to the training boundaries which are zones of poor accuracy for standard PINNs and results in a noticeable global accuracy improvement at iso-budget. Finally, we propose for the first time to relax the incompressibility condition in such a way that it drastically benefits the optimization search and results in a much improved convergence of the composite loss function. The RB results obtained at high Rayleigh number Ra = 2 $\bullet$ 10 9 are particularly impressive: the predictive accuracy of the surrogate over the entire half a billion DNS coordinates yields errors for all flow variables ranging between [0.3% -- 4%] in the relative L 2 norm, with a training relying only on 1.6% of the DNS data points.

Motivation & Objective

• Assess the feasibility of PINNs as surrogates for 3D turbulent natural convection modeled by NS with Boussinesq approximation.
• Quantify data requirements and residual-sampling strategies necessary to accurately recover flow fields from partial DNS data.
• Introduce a padding technique to distribute residuals around labeled data regions to regularize training.
• Propose relaxing the incompressibility constraint to enhance optimization convergence and surrogate accuracy.

Proposed method

• Formulate the unsteady Navier–Stokes equations under the Boussinesq approximation within a PINN framework.
• Use a multi-layer perceptron to approximate the solution vector u = (v, p, T) and compute PDE residuals via automatic differentiation.
• Train with a loss combining labeled data and PDE residuals, using Adam with staged learning-rate cycles.
• Introduce a padding strategy that distributes residual evaluation points around regions with labeled data to improve boundary accuracy.
• Add an auxiliary transported temperature variable T̄ = 1 − T to provide additional training constraints.
• Relax the incompressibility condition to improve optimization search and convergence of the composite loss.

Experimental results

Research questions

• RQ1 Can PINNs accurately surrogate 3D turbulent natural convection at high Rayleigh numbers with partial DNS data?
• RQ2 What data and residual-sampling strategies best balance labeled data and PDE residuals for training PINNs on turbulent RB convection?
• RQ3 Does a padding scheme around labeled data enhance PINN accuracy, particularly near training boundaries?
• RQ4 Does relaxing the incompressibility constraint improve optimization convergence and surrogate performance?

Key findings

• Achieved relative L2 errors across flow variables in the range of 0.3% to 4% for high-Ra RB convection when using only 1.6% of DNS points for training.
• Demonstrated that PDE residuals dominate early in training, while data-label errors converge more regularly with cycles.
• Introduced a padding technique that distributes residual points around labeled data regions, yielding notable global accuracy improvements at iso-budget.
• Relaxing the incompressibility constraint substantially improves optimization search and convergence of the PINN loss.
• Under Ra = 2×10^9, the surrogate trained on a fraction of DNS data can predict the full field over a half-billion DNS coordinates with high fidelity.