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

Area of Science:

• Computational Mathematics
• Applied Physics
• Machine Learning

Background:

• High-dimensional nonlinear partial differential equations (PDEs) like the sine-Gordon equation (SGE) pose significant numerical challenges.
• Standard physics-informed neural networks (PINNs) struggle with accuracy and stability in high dimensions and complex scenarios.

Purpose of the Study:

• To develop a novel physics-informed neural network (PINN) framework, termed Adaptive Weighted Loss Gradient-Enhanced PINNs (AWL-gPINNs).
• To enhance the numerical approximation of high-dimensional nonlinear sine-Gordon equations (SGEs).
• To improve upon the accuracy, convergence, and robustness of existing numerical methods for PDEs.

Main Methods:

• Modified PINN formulation incorporating gradient-based residual constraints and an adaptive weighting strategy.
• Reduction of the SGE to a coupled first-order form to facilitate automatic differentiation.
• Development of a composite loss functional with trainable weights for PDE residuals and gradient regularization.

Main Results:

• AWL-gPINNs achieved significant error reductions (orders of 1e-3 to 1e-5) compared to standard PINNs and other algorithms.
• Demonstrated rapid convergence, enhanced training robustness, and stability across various conditions (dimensionality, noise, initializations).
• Outperformed state-of-the-art PINN variants in high-dimensional PDE approximations, including 20D cases.

Conclusions:

• AWL-gPINNs offer a scalable and effective technique for solving high-dimensional nonlinear PDEs.
• The adaptive weighting and gradient enhancement strategies are crucial for overcoming stiffness and multi-objective optimization imbalances.
• The proposed framework shows superior performance and practical efficiency for complex scientific computations.