2026/07/01 by Salah A. Faroughi, Farinaz Mostajeran, Amirhossein Arzani +1
Computer Science · #Explainable Artificial Intelligence (XAI) #Advanced Graph Neural Networks #Machine Learning and Algorithms
paper · doi:10.1016/j.jcp.2026.115223
Symbolic discovery of governing equations is a long-standing goal in scientific machine learning, yet a fundamental trade-off remains between interpretability and learning efficiency. Classical symbolic regression methods yield explicit analytic expressions but rely on combinatorial search, whereas neural networks scale efficiently with data and dimensionality but produce opaque representations. We introduce Symbolic Kolmogorov-Arnold Networks (Symbolic-KANs), a neural architecture that embeds symbolic structure directly within a trainable network. Symbolic-KANs represent multivariate functions as compositions of learned univariate primitives applied to learned scalar projections, guided by a library of analytic functions, hierarchical gating, and symbolic regularization that progressively sharpens continuous mixtures into one-hot selections. After training, each active unit selects a single primitive and projection direction, producing compact closed-form expressions without post-hoc symbolic fitting. Beyond symbolic regression, Symbolic-KAN serves as an efficient primitive-discovery mechanism that can inform candidate libraries for sparse equation-learning methods. We demonstrate accurate recovery of governing structures in data-driven regression and inverse dynamical systems, and extend the framework to forward and inverse physics-informed learning of partial differential equations (PDEs). We further show robustness to noisy boundary measurements in elliptic PDEs and applicability to nonlocal Volterra integro-differential equations, where Symbolic-KAN maintains accurate predictions while identifying primitives consistent with the underlying analytical structure. These results establish Symbolic-KAN as a practical and interpretable framework for learning governing laws from data and physics.