Bayesian hyper-parameter optimization of physics-informed neural networks to a nonlocal nonlinear Schrödinger equation

In previous applications of a physical information neural networks (PINN) algorithm, the adjustment of hyper-parameter combinations for neural networks has predominantly been conducted manually. This manual configuration strategy often suffers from inefficiency, and it cannot guarantee that the network trained with such parameters can achieve optimal performance. To address the hyper-parameter tuning challenge of PINNs, this paper is devoted to introducing a Bayesian hyper-parameter optimization method for PINN (BHPO-PINN) based on a Gaussian process. This approach enhances training performance by identifying optimal combinations of different hyper-parameters and is further applied to solve the nonlocal reverse-time nonlinear Schrödinger equation. Numerical simulations of a data-driven one-soliton solution and four types of two-soliton solutions are performed, and systematic comparative analyses of relative errors and maximum absolute errors are presented to quantify the solution accuracy. To identify the optimal combination of network hyper-parameters, we conduct a thorough investigation by analyzing error variations under different training iterations. The experimental results demonstrate that the BHPO-PINN method achieves significantly enhanced accuracy and stability compared to the traditional PINN algorithm under empirically configured hyper-parameters.