Gradient-optimized physics-informed neural networks (GOPINNs): a deep learning method for solving the complex modified KdV equation

Recently, the physics-informed neural networks (PINNs) have received more and more attention because of their ability to solve nonlinear partial differential equations via only a small amount of data to quickly obtain data-driven solutions with high accuracy. However, despite their remarkable promise in the early stage, their unbalanced back-propagation gradient calculation leads to drastic oscillations in the gradient value during model training, which is prone to unstable prediction accuracy. Based on this, we develop a gradient optimization algorithm, which proposes a new neural network structure and balances the interaction between different terms in the loss function during model training by means of gradient statistics, so that the newly proposed network architecture is more robust to gradient fluctuations. In this paper, we take the complex modified KdV equation as an example and use the gradient-optimized PINNs (GOPINNs) deep learning method to obtain data-driven rational wave solution and soliton molecules solution. Numerical results show that the GOPINNs method effectively smooths the gradient fluctuations and reproduces the dynamic behavior of these data-driven solutions better than the original PINNs method. In summary, our work provides new insights for optimizing the learning performance of neural networks and improves the prediction accuracy by a factor of 10 to 30 when solving the complex modified KdV equation.