Learning forward and inverse dynamics of nonlinear complex-valued partial differential equations using Kolmogorov–Arnold networks

Abstract

In recent years, we have witnessed the application of a number of neural networks to the numerical solution of some partial differential equations (PDEs). In the present work, we present Kolmogorov–Arnold networks and their applications for complex-valued nonlinear PDEs (CNPDEs). These networks have demonstrated great potential for data-driven modelling, as an alternative to multilayer perceptrons (MLPs). We propose a new method called Complex Physics-Informed Kolmogorov–Arnold Network (CPIKAN) combining the learning capabilities of KANs with the rigour of physics equations. By directly integrating the equation into the network loss function, we can capture the complexity of nonlinear models and provide accurate numerical solutions. We evaluate the performance of CPIKAN by solving direct and inverse complex-valued problems, and compare the obtained results with analytical results. Our results demonstrate the potential of CPIKAN for solving complex-valued PDEs, opening new perspectives for the modelling and simulation of complex systems.