A gradient-enhanced physics-informed neural network with adaptive loss weighting for high-dimensional non-linear sine-Gordon problems.

Abstract

This paper proposes a modified physics-informed neural network (PINN), known as Adaptive Weighted Loss Gradient-Enhanced PINNs (AWL-gPINNs) to the numerical approximation of high-dimensional nonlinear sine-Gordon equations (SGEs). The proposed algorithm is an extension of the typical PINN formulation, i.e. it adds gradient-based residual constraints and an adaptive weighting strategy in order to balance the importance of PDE residual terms, initial conditions, and boundary conditions in training. The existing governing SGE is reduced to a coupled first-order form, which allows the automatic differentiation to be easily integrated to evaluate higher-order derivatives. The resulting composite loss functional consists of the residual and gradient-regularization terms that have trainable weights, which reduce the issue of stiffness and imbalance in multi-objective optimization. Benchmark problems of 3D to 20D damped and undamped SGEs in both long and short time domains are extensively numerically experimented with. The findings show that AWL-gPINNs perform much better than standard PINNs and a variety of existing algorithms and obtain orders of error reduction between 1e-3 -1e-2 and 1e-5-1e-4. The technique also demonstrates rapid convergence, increased training robustness, and stability across different collocation densities, noise perturbations, and initialization conditions. Moreover, when the proposed method is compared to state-of-the-art variants of PINN, it is established that the method is superior to the current methods in a variety of high-dimensional PDEs with very small error magnitudes, even in the 20D case. The efficacy, robustness, and practical efficiency of the suggested AWL-gPINN framework for high-dimensional nonlinear SGE are further validated by ablation studies and multi-seed stability assessments. The results support that the proposed AWL-gPINNs is a scalable and successful technique for high-dimensional nonlinear PDE solutions.