Physics Informed Neural Networks for Modeling of 3D Flow-Thermal Problems with Sparse Domain Data

TL;DR

The paper demonstrates that Physics Informed Neural Networks (PINNs) can solve 3D, highly nonlinear Navier–Stokes problems on complex geometries using extremely sparse domain data, by leveraging a hybrid data-physics training strategy and Fourier feature embeddings to enhance learning. It shows through stenosis and PCB flow experiments that incorporating physics into the loss improves convergence and solution quality compared with purely data-driven models, while recognizing data sparsity can still yield unphysical results if data are too scarce. The study extends PINN-based surrogate modeling to a heat-sink design problem, where a hybrid PINN surrogate coupled with a PSO-based optimizer achieves design points that CFD validates with substantial speedups over traditional CFD-driven optimization. Collectively, these results highlight PINNs’ potential for rapid 3D flow-thermal simulations and design optimization, while outlining avenues for improved convergence guarantees, turbulence modeling, uncertainty quantification, and geometry-generalizable surrogates.

Abstract

Successfully training Physics Informed Neural Networks (PINNs) for highly nonlinear PDEs on complex 3D domains remains a challenging task. In this paper, PINNs are employed to solve the 3D incompressible Navier-Stokes (NS) equations at moderate to high Reynolds numbers for complex geometries. The presented method utilizes very sparsely distributed solution data in the domain. A detailed investigation on the effect of the amount of supplied data and the PDE-based regularizers is presented. Additionally, a hybrid data-PINNs approach is used to generate a surrogate model of a realistic flow-thermal electronics design problem. This surrogate model provides near real-time sampling and was found to outperform standard data-driven neural networks when tested on unseen query points. The findings of the paper show how PINNs can be effective when used in conjunction with sparse data for solving 3D nonlinear PDEs or for surrogate modeling of design spaces governed by them.

Figures (21)

• Figure 1: The spectrum of data-driven versus physics-informed models. Incorporating governing physics information into the models during creation serves as an effective form of regularization and often helps reduce the amount of data required to achieve the same accuracy levels.
• Figure 2: Adaptive coefficients and loss terms from Equation \ref{['eqn:lr_annealing_loss']} during training. (\ref{['fig:modded_lr_dir_const']}) Evolution of the Dirichlet loss adaptive constant during training.(\ref{['fig:modded_lr_neu_const']}) Evolution of the Neumann loss adaptive constant during training. (\ref{['fig:modded_loss_dir_loss']}) Dirichlet B.C loss term $L_{dir}(\theta)$ (\ref{['fig:modded_loss_neu_loss']}) Neumann B.C loss term $L_{neu}(\theta)$ (\ref{['fig:modded_loss_pde_loss']}) The PDE loss during training. Once the values of both the adaptive constants start dropping, the PDE loss improves much more rapidly.
• Figure 3: Visual description of stenosis problem
• Figure 4: Stenosis diagram (not to scale) showing planes where solution data is provided randomly.
• Figure 5: Solution Comparison. (\ref{['fig: acuSolve_stenosis_contours']}) Altair AcuSolve® Solution to stenosis problem (\ref{['fig:PINN_stenosis_contours']}) PINN forward solve to stenosis problem.