NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elements

TL;DR

NeuroSEM is a hybrid framework integrating physics-informed neural networks with the high-fidelity Spectral Element Method (SEM) solver, Nektar++ that leverages the strengths of both PINNs and SEM, providing robust solutions for multiphysics problems.

Abstract

Multiphysics problems that are characterized by complex interactions among fluid dynamics, heat transfer, structural mechanics, and electromagnetics, are inherently challenging due to their coupled nature. While experimental data on certain state variables may be available, integrating these data with numerical solvers remains a significant challenge. Physics-informed neural networks (PINNs) have shown promising results in various engineering disciplines, particularly in handling noisy data and solving inverse problems in partial differential equations (PDEs). However, their effectiveness in forecasting nonlinear phenomena in multiphysics regimes, particularly involving turbulence, is yet to be fully established. This study introduces NeuroSEM, a hybrid framework integrating PINNs with the high-fidelity Spectral Element Method (SEM) solver, Nektar++. NeuroSEM leverages the strengths of both PINNs and SEM, providing robust solutions for multiphysics problems. PINNs are trained to assimilate data and model physical phenomena in specific subdomains, which are then integrated into the Nektar++ solver. We demonstrate the efficiency and accuracy of NeuroSEM for thermal convection in cavity flow and flow past a cylinder. We applied NeuroSEM to the Rayleigh-Bénard convection system, including cases with missing thermal boundary conditions and noisy datasets, and to real particle image velocimetry (PIV) data to capture flow patterns characterized by horseshoe vortical structures. The framework's plug-and-play nature facilitates its extension to other multiphysics or multiscale problems. Furthermore, NeuroSEM is optimized for efficient execution on emerging integrated GPU-CPU architectures. This hybrid approach enhances the accuracy and efficiency of simulations, making it a powerful tool for tackling complex engineering challenges in various scientific domains.

Figures (18)

• Figure 1: Coupling $\text{PINNs}$ and $\text{X}$, where $\text{X}$ can be any existing numerical methods, e.g. SEM, FVM and FDM. In this study, we choose SEM as the backbone and propose the NeuroSEM method to solve the multiphysics problem.
• Figure 2: Case A: Integration of PINNs for $T$ in Nektar++. The algorithm presented here to solve Eqs. \ref{['eq:problem_a']} and \ref{['eq:problem_b']} is motivated by the work of karniadakis1991high.
• Figure 3: Case (B): Integration of the PINNs model for $\bm{u}$ in Nektar++. The integration computes $T$ by solving Eq. \ref{['eq:problem_c']}, which is a linear advection PDE. The advection velocity $\bm{u}$ in Eq. \ref{['eq:problem_c']} is computed by the pre-trained PINNs model.
• Figure 4: Problem setups for the Rayleigh-Bénard convection \ref{['eq:problem']} with buoyancy driven thermal convection.
• Figure 5: Scenario A: Streamline plots obtained from NeuroSEM (top row) and SEM (bottom row). The left, center and right subfigures show the streamline plots for $\text{Ra}=10^4$, $10^5$ and $10^6$, respectively. The errors of $\bm{u} = [u, v]^T$ are presented in Table \ref{['tab:scenario_a']}.