Data-driven soliton mappings for the fifth-order Caudrey–Dodd–Gibbon equation via the HyperFNO model

Abstract In this paper, we extend the Fourier neural operator (FNO) to explore its application in discovering mappings for soliton solutions of the fifth-order Caudrey–Dodd–Gibbon (CDG) equation. We implement an enhanced model—hypernetworks for FNO (HyperFNO)—and employ it to predict mappings for soliton solutions. To comprehensively evaluate the model’s performance, we record loss data during both the training and testing processes, comparing the data-driven soliton solutions with theoretical exact solutions to verify the model’s accuracy. Notably, the losses achieved by our models are on the order of O(10−7). We systematically investigate the effect of various key factors on the performance of the HyperFNO algorithm, including different activation functions (such as ReLU, Sigmoid, ELU and tanh) as well as various loss functions and optimizers. In addition, we explore the effect of Gaussian perturbations on the solutions of the equation. Finally, we demonstrate the performance advantages of HyperFNO by comparing HyperFNO with FNO. The results of this study not only validate the effectiveness of HyperFNO in handling complex-equation solution mappings but also provide significant insights into the potential of this neural network for spatial mapping problems.