LoKAN: Back prop-Free Node-Local Functional-Edge Learning for Equation Discovery

Kolmogorov-Arnold network (KAN) model nonlinear relationships through functions on edges, but their many learnable edge parameters typically require repeated updates through backpropagation, making training costly. Such iterative optimization does not directly exploit the linear structure of edge coefficients under fixed basis functions to improve training efficiency. Fitting edges independently, in turn, ignores the coupling among incoming edges of the same node and generally cannot guarantee the joint optimum for a fixed support. To address these limitations, we propose LoKAN, a single-layer functional-edge model for equation discovery. Given task-specified candidate features, LoKAN constructs anchored piecewise-linear spline edges, jointly solves a ridge-regression system over the active incoming edges of each output node, iteratively prunes edges by their normalized contributions, and refits on the final support, without backpropagating through edge coefficients. In addition, LoKAN uses exact Gram-component partitioning to perform component-wise solves, reducing matrix storage requirements. Separate efficiency and scaling experiments show, respectively, that LoKAN achieves shorter per-fit training times than the compared backpropagation-trained KAN and spline group lasso, and that component-wise solves reduce Gram-matrix storage. Across the experiments, LoKAN demonstrates strong predictive performance and effectively recovers explicit candidate terms.