Physics-informed deep learning for incompressible laminar flows

Abstract: Physics-informed deep learning has drawn tremendous interest in recent years to solve computational physics problems, whose basic concept is to embed physical laws to constrain/inform neural networks, with the need of less data for training a reliable model. This can be achieved by incorporating the residual of physics equations into the loss function. Through minimizing the loss function, the network could approximate the solution. In this paper, we propose a mixed-variable scheme of physics-informed neural network (PINN) for fluid dynamics and apply it to simulate steady and transient laminar flows at low Reynolds numbers. A parametric study indicates that the mixed-variable scheme can improve the PINN trainability and the solution accuracy. The predicted velocity and pressure fields by the proposed PINN approach are also compared with the reference numerical solutions. Simulation results demonstrate great potential of the proposed PINN for fluid flow simulation with a high accuracy.

• The paper presents a mixed-variable PINN method that reformulates the Navier-Stokes equations to enhance neural network trainability.
• It demonstrates superior performance over traditional PINNs in steady and transient flow scenarios with reduced relative ℓ2 errors.
• The approach leverages physics-based constraints to simulate incompressible laminar flows efficiently, offering a data-efficient alternative for computational fluid dynamics.

An Exploration of Physics-Informed Deep Learning for Incompressible Laminar Flows

This paper presents a study on the application of physics-informed neural networks (PINNs) to simulate incompressible laminar flows, focusing on exploiting physical laws to constrain neural network training. The authors introduce a mixed-variable PINN for fluid dynamics, emphasizing its efficacy in improving trainability and solution accuracy over traditional schemes.

Methodology Overview

The proposed methodology leverages the principles of physics-informed deep learning, which seeks to incorporate the residuals of the governing physics equations into the loss function of a neural network. Unlike conventional data-driven approaches, where substantial data is required to train models, PINNs demand less data by embedding physical knowledge—thus reducing dependency on large datasets.

In this study, the authors address the challenges of PINN applicability to fluid dynamics by using a mixed-variable scheme. They start with a reformulation of the Navier-Stokes equations into more tractable continuum and constitutive law forms. This reformulation reduces the order of derivatives required in PINN training, thus enhancing trainability. Moreover, the stream function is used instead of direct velocity terms to inherently satisfy the divergence-free condition for incompressible flows.

Results and Comparative Analysis

The authors implemented their mixed-variable PINN approach to model both steady and transient flow scenarios around a circular cylinder. A key comparative study evaluated the efficacy of the mixed-variable PINN against traditional PINNs. The results indicate that the mixed-variable scheme exhibits superior performance in both trainability and accuracy. The predicted velocity and pressure fields show strong agreement with reference solutions obtained via ANSYS Fluent simulations.

For the steady flow case, the mixed-variable approach significantly reduced relative $\ell_2$ errors in velocity fields compared to traditional methods. The parameter study on the loss function coefficients further demonstrated the proposed scheme’s robustness to variations in hyperparameter settings, a property less prominent in the traditional PINN approach.

The transient flow study illustrated the capacity of the proposed method to effectively capture time-evolving dynamics of fluid flows without any simulation or real-world measurement data. The temporal evolution of pressure fields in this scenario also matched closely with benchmark solutions, showcasing the versatility and applicability of the mixed-variable PINN in processing time-dependent flow challenges.

Implications and Future Directions

The implications of this research are extensive. The proposed PINN framework could significantly impact the computational physics community by providing a more data-efficient, physically-consistent alternative to traditional numerical solvers. It presents potential advantages in engineering fields where flow simulation with limited data is necessary.

Despite the promising results, the authors acknowledge limitations when applying this approach to high Reynolds number or turbulent flows. The computational demands remain a challenge due to the need for finer discretization in such scenarios. Future research directions include the development of hybrid strategies combining PINNs with advanced methods, such as transfer learning, to manage these demands more effectively. The proposed "divide-and-conquer" strategy through transfer learning could provide a pathway for handling large-scale, complex flow problems within the PINN framework.

In conclusion, this paper contributes a significant advancement in the application of PINNs for fluid dynamics, particularly in improving neural network trainability and prediction accuracy for incompressible laminar flows. The study opens new avenues for research in physics-informed deep learning with promising potential for applications across computational mechanics and engineering disciplines.