Energy-dissipative evolutionary Kolmogorov-Arnold networks for complex PDE systems

Abstract

In this work, we introduce an evolutionary Kolmogorov-Arnold Network (EvoKAN), a novel framework for solving complex partial differential equations (PDEs). EvoKAN builds on Kolmogorov-Arnold Networks (KANs), where activation functions are spline-based and trainable on each edge, offering localized flexibility across multiple scales. Rather than retraining the network repeatedly, EvoKAN encodes only the initial state of the PDE during an initial learning phase. EvoKAN models Kolmogorov-Arnold network weights as time-dependent functions and updates them through the evolution of the governing PDEs. By treating EvoKAN weights as continuous functions in the relevant coordinates and updating them over time, EvoKAN can predict system trajectories over arbitrarily long horizons, a notable challenge for many conventional neural network-based methods. In addition, EvoKAN employs the scalar auxiliary variable (SAV) method to guarantee unconditional energy stability and computational efficiency. At individual time step, SAV only needs to solve the decoupled linear systems with constant coefficients, the implementation is significantly simplified. We test the proposed framework in several complex PDEs, including one-dimensional and two-dimensional Allen–Cahn equations and two-dimensional Navier-Stokes equations. The numerical results show that the EvoKAN solutions closely match the analytical references and established numerical benchmarks, effectively capturing the sharp interfaces in predicting the solution of the Allen-Cahn equation and turbulent flow patterns in predicting the solution of the Navier-Stokes equations.