Extraction and reconstruction of variable-coefficient governing equations using Res-KAN integrating sparse regression

Extracting the governing equations for complex systems plays a crucial role in scientific discovery and engineering applications. Previous research often focus on static properties of governing equations, while real-world dynamics involve complex, evolving factors that influence the system behavior. This work proposes a novel approach that integrates the single-layer Kolmogorov-Arnold networks (KANs) in the downstream operations of physics-informed neural networks (PINNs), combined with an alternating training strategy using sparse regression algorithms. Different from the traditional methods, this approach relies solely on sparse data, without any prior knowledge, to reconstruct precise form of governing equations and simultaneously identify the variable-coefficient functions depending on single variables. By symbolizing the spline functions in the KAN layer, it can also derive the exact expressions of these coefficient functions and reveal the key parameters of real physical significance. Furthermore, when extending the framework to high-dimensional problems, KAN’s regularization enables weight sparsity enforcement, which removes redundant neurons and optimizes the network. Experiments on various benchmark problems demonstrate the robustness of our method to varying levels of data sparsity and noise, offering a new solution to the reconstruction and analysis of the governing equations.