Reconstructing higher-order constitutive equations based on deep learning

To overcome the problems of high computational complexity, low efficiency, and limited accuracy in the iterative solution of traditional high-order constitutive equations, this paper proposes a deep learning-based high-order constitutive equation reconstruction (DLHCER) method. The method embeds the nonlinear terms into the source terms of the Navier–Stokes (NS) equations, and the DLHCER control equations are obtained by physical reconstruction. In the training stage, the error between the experimental data and computational fluid dynamics simulation data is utilized to generate the training data and establish the mapping relationship between the free incoming flow variables and the characteristic nonlinear variables. Ultimately, the complex nonlinear terms in DLHCER are fitted and predicted by an agent model to capture the complex relationship between the source term parameters and the incoming flow conditions, thus replacing the traditional iterative solution process. Experimental results show that the average relative error between the theoretical and predicted values of the DLHCER model is 0.302%. The agent model is further analyzed for Shapley additive explanations interpretability to clarify the relationship between network inputs and outputs. Through the comparative validation of nonlinear coupled constitutive relations, Navier–Stokes–Fourier and DLHCER on one-dimensional positive excitation wave arithmetic cases, the results show that DLHCER significantly simplifies the computational process by directly correcting the free parameters of the source term, avoids the cumbersome iterative process, and realizes the double enhancement of computational efficiency and accuracy. Therefore, DLHCER is expected to provide strong support for higher-order constitutive equations in engineering applications in the field of dilute gases.