In this paper, we employ Deep Operator Networks (DeepONets) to solve a class of higher-order nonlinear PDEs. We focus on the fifth-order Lax equation, which features fifth-order spatial derivatives, strong nonlinearity, and localized soliton structures. We also study the generalized two-dimensional equation and the two-dimensional Burgers equation, both of which involve multi-dimensional coupling and complex evolution behaviors. Our results demonstrate that DeepONet effectively handles these complexities, achieving a mean squared error (MSE) of \(\mathcal {O}(10^{-7})\) and a discrete relative \(L^2\) error of \(\mathcal {O}(10^{-3})\) . We analyze the impact of activation functions, initializers, and input function spaces, and compare DeepONet with FNN and ResNet architectures. DeepONet exhibits strong robustness against Gaussian perturbations of \(1\%\) and \(5\%\) , maintaining an MSE of \(\mathcal {O}(10^{-5})\) and a discrete relative \(L^2\) error of \(\mathcal {O}(10^{-3})\) . When applied to two higher-dimensional nonlinear PDEs, it maintains consistent performance with an MSE of \(\mathcal {O}(10^{-5})\) and a discrete relative \(L^2\) error of \(\mathcal {O}(10^{-3})\) . These numerical experiments demonstrate DeepONet’s effectiveness in solving soliton solutions of the fifth-order Lax equation and its ability to capture the intricate dynamics of higher-dimensional nonlinear PDEs with remarkable precision and stability.
