Network-Enhanced Gaussian Process-Assisted Multi-Objective Parameter Optimization of Hybrid WENO Schemes for the Compressible Euler Equations

In hybrid weighted essentially non-oscillatory (WENO) schemes, the coupled effects of smoothness detection, nonlinear weighting, and dissipation control give rise to complex nonlinear responses to parameter changes. This coupling makes it challenging to improve smooth-region accuracy and shock resolution simultaneously. Moreover, the high cost of computational fluid dynamics (CFD) evaluations limits the efficiency of direct multi-objective searches. To address these challenges, we couple a network-enhanced Gaussian process model (NEGPM) with the non-dominated sorting genetic algorithm II (NSGA-II) to optimize hybrid WENO parameters for the compressible Euler equations. NEGPM uses a shared neural encoder supervised by both objectives to capture nonlinear interactions among continuous and discrete parameters. In the learned latent space, two independent Gaussian processes with Matérn-5/2 kernels and automatic relevance determination predict smooth-region error and numerical shock thickness. A two-stage training strategy combines shared-feature learning with kernel hyperparameter estimation via marginal likelihood, enabling target-specific predictions from limited CFD samples. These predictions guide non-dominated search, with the selected design subsequently verified by CFD. The model achieves log-space R2 values of 0.9957 and 0.9853 for the two objectives, respectively, and corresponding Spearman coefficients of 0.9985 and 0.9935. This accuracy and ranking consistency support its use in multi-objective optimization. On the Euler benchmarks considered, the optimized scheme reduces smooth-region error by 27.91% and numerical shock thickness by 10.78% relative to WENO-Z. Moreover, the framework achieves optimized performance comparable to direct CFD-driven NSGA-II while reducing CFD evaluations from 2000 to 97. Total optimization time decreases from 126 h 34 min to 6 h 18 min. The framework thus offers an efficient, data-driven approach to improving numerical performance in high-order shock-capturing scheme design.