Error estimates for a physics-informed neural network in solving KdV equations

This paper aims to provide error bounds on physics-informed neural network (PINN) in solving Korteweg–de Vries (KdV) equations. We prove that a neural network equipped with two hidden layers and the tanh activation function can reduce the partial differential equation residuals arbitrarily. The generalization error and training error can be bounded by the number of training points and the width of the aforementioned neural network. Besides, the upper bound of the total error can be controlled by the generalization error. These error bounds offer a theoretical understanding of PINN’s ability in solving KdV equations. A series of parameterized KdV equations are also conducted to demonstrate the performance of PINN when solving KdV equations.