Energy-Dissipative Evolutionary Kolmogorov-Arnold Networks for Complex PDE Systems

TL;DR

EvoKAN introduces a physics-informed framework that evolves Kolmogorov–Arnol’d Networks (KANs) over time to solve time-dependent PDEs without recurrent retraining. By combining edge-based spline activations, time-varying network parameters via an evolutionary PDE-driven update, and the Scalar Auxiliary Variable (SAV) method for unconditional energy stability, EvoKAN achieves accurate long-horizon predictions for challenging problems like Allen–Cahn and Navier–Stokes equations. The approach yields high fidelity against analytical and spectral references, and demonstrates superior stability and robustness compared to evolutionary neural networks on turbulent-like NSE regimes. This offers a scalable, energy-stable, and flexible tool for simulating complex PDE systems with potential for real-time and multiscale applications.

Abstract

We introduce evolutionary Kolmogorov-Arnold Networks (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 PDE's initial state during an initial learning phase. The network parameters then evolve numerically, governed by the same PDE, without any additional optimization. By treating these parameters as continuous functions in the relevant coordinates and updating them through time steps, EvoKAN can predict system trajectories over arbitrarily long horizons, a notable challenge for many conventional neural-network based methods. In addition, EvoKAN integrates the scalar auxiliary variable (SAV) method to guarantee unconditional energy stability and computational efficiency. At individual time step, SAV only needs to solve 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. Numerical results show that EvoKAN solutions closely match analytical references and established numerical benchmarks, effectively capturing both phase-field phenomena (Allen-Cahn) and turbulent flows (Navier-Stokes).

Figures (6)

• Figure 1: Illustration of Evolutionary Kolmogorov–Arnold Networks (EvoKAN). Kolmogorov–Arnold Networks (KANs) replace linear weights with univariate spline functions learned on each edge. In the evolutionary framework, EvoKAN models Kolmogorov–Arnold Network weights as time-dependent functions and updates them through the evolution of the governing PDEs—without additional training of KAN.
• Figure 2: Comparative analysis of benchmark solution and EvoKAN solution for one-dimensional Allen–Cahn equation. The three panels display the benchmark solution (top), EvoKAN solution (middle), and EDNN solution (bottom).
• Figure 3: Comparative analysis of benchmark solution, EvoKAN, and EDNN solutions for the two-dimensional Allen–Cahn equation under different initial conditions. Panels (a) and (b) show the results for initial conditions with $\alpha = 1$ and $\alpha = 2$, respectively.
• Figure 4: Initial conditions for the two-dimensional Navier-Stokes equations. The figure presents the initial velocity fields ($u$) (first column) and ($v$) (second column), pressure field ($P$) (third column), and vorticity ($\omega$) (fourth column).
• Figure 5: Comparison of benchmark and EvoKAN for two-dimension NSE with $\nu=0.05$: Velocity fields, $u$ (first column) and $v$ (second column) , pressure $P$ (third column) , and vorticity $\omega$ (fourth column).