(226f) Physics-Informed Neural Network for Solving 2-D Heat Transfer Problems without Labeled Data

The accurate simulation of the basic heat transfer behavior is an indispensable step in designing high-efficient and low-cost heat transfer equipment such as coolers, heaters, and heat exchangers. Generally, the heat transfer system can be represented by a highly nonlinear partial differential equation (PDE) system that is governed by the Navier–Stokes equations. This numerical representation of the heat transfer system primarily relies on solving the PDE systems via traditional mesh-based methods such as the finite difference method and the finite element method, which is also known as the computational fluid dynamics (CFD). In practice, however, CFD simulations are often computationally challenging, especially for heat exchangers with complex turbulence flows and enclosed geometries. Besides, it is hard to integrate the principled CFD models with the flow process simulation model or the mathematical programming model, which thus greatly limits its further application to many-query analysis and real-time predictions.

In recent years, the mesh-free deep learning method shows great potential to deal with high-dimensional PDE problems and could overcome the curse of dimensionality in certain problems by taking advantage of the automatic differentiation^[1]. Among them, the physics-informed neural network (PINN) proposed by Raissi et al.^[2] was used to directly integrate the PDE in a strong form, which can encode the physical conservation laws and prior physical knowledge (e.g. boundary conditions) into the neural network. In this way, the PINN can significantly reduce the dependence on labeled input-output data for the traditional data-driven modeling and get rid of the limits of expensive CFD modeling^[3]. Note that, the training effect of a PINN is closely related to the treatment of boundary conditions. Previously, the outputs of the deep neural network are directly used to construct the PDE residuals in the loss function via automatic differentiation. The boundary conditions imposed in PINN as constraints are often treated in a “soft” manner by modifying the original loss function with penalty terms. However, this soft PINN treatment is difficult to guarantee the solution accuracy of the boundary conditions being imposed. To overcome this drawback, it is proposed that the boundary conditions in PINN can be imposed in a “hard” manner^[4,5]. As shown in Fig. 1, the boundary conditions in the modified architecture are encoded with the output of the deep neural network to form the output of the hard PINN before automatic differentiation. In this way, only the PDE residuals are included in the loss function and the boundary constraints can be automatically fulfilled, which also accordingly accelerates the convergence rate.

In this work, the PINNs are trained to construct the surrogate models of the temperature fields of heat transfer for given input vectors (i.e., space coordinates and parameters of the PDEs). Two 2-D steady-state heat transfer systems with an internal heat source, namely heat conduction model and convection heat transfer between plates are taken as illustrative examples. In the first example, Fig. 2 shows that the temperature fields predicted by soft PINN and hard PINN are comparable with the CFD solutions. However, the prediction ability of hard PINN is superior to the rival due to the lower maximum relative errors (soft PINN 0.60%, hard PINN: 0.015%). In the second example, the hard PINN is further to predict the behavior of the convection heat transfer surrogate model in the plate heat exchanger. As shown in Fig. 3, the prediction results of the constructed surrogate model match well with the CFD solution. The maximum absolute error of the temperature field is only 0.0038 ^oC. It can be foreseen that the PINN would open up a new way for constructing high-fidelity surrogate models.

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[2] Raissi M, Perdikaris P, Karniadakis G E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational Physics, 2019, 378: 686-707.

[3] Zhu Y, Zabaras N, Koutsourelakis P S, et al. Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data[J]. Journal of Computational Physics, 2019, 394: 56-81.

[4] Sun L, Gao H, Pan S, et al. Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 361: 112732.

[5] LU Zhibin, QU Jinghui, LIU Hua, HE Chang, ZHANG Bingjian, CHEN Qinglin. Surrogate modeling for physical fields of heat transfer processes based on physical information neural network[J]. CIESC Journal, 2021, 72(3): 1496-1503.