Adaptive deep physics-informed neural network with dual-nested activation for solving complex partial differential equations

Physics-informed neural networks (PINNs) hold promise for solving partial differential equations (PDEs), but they often face challenges in achieving high accuracy, especially in complex, real-world scenarios. This paper presents an adaptive deep PINN (ad-PINN) framework designed to enhance the efficiency of both activation and loss functions. The proposed ad-PINN introduces two main innovations: (1) an adaptive activation function with dual-nested mechanism, called the dual-tanh function, which dynamically adjusts its slope and shape to optimize learning capacity beyond traditional activations; and (2) an adaptive Huber loss function, which automatically adjusts its parameters, eliminating the need for manual tuning. This dual adaptability in activation and loss functions improves the model’s flexibility and performance. Theoretically, we demonstrate that with proper initialization and learning rates, a gradient descent algorithm minimizing the loss function avoids convergence to suboptimal points or local minima. The effectiveness of ad-PINN is showcased through real-world applications, including shockwave propagation and reflection, incompressible solid mechanics, bi-material problems, and fluid dynamics. Comparative experiments reveal that ad-PINN achieves significantly higher accuracy than some existing PINNs, highlighting its capability to tackle complex problems with high-gradient solutions, capture hidden incompressibility, handle displacement discontinuities, manage high-dimensional systems with complex geometric and physical features, and address systems governed by partially known physical laws.