arXiv 2603.238542026-03-25cs.LGcs.CEmath.APmath.DS

Symbolic-KAN：带有离散符号结构的Kolmogorov-Arnold网络用于可解释学习

Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning

Energy \& Intelligence Lab, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA；Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA；Imaging Institute, University of Utah, Salt Lake City, Utah, 84112, USA；Department of Biomedical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA；Department of Mechanical Engineering, School of Engineering, Urmia University of Technology, Urmia, Iran

Salah A Faroughi, Farinaz Mostajeran, Amirhossein Arzani, Shirko Faroughi

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AI总结：

本文提出Symbolic-KAN，通过在可训练深度网络中嵌入离散符号结构，解决可解释性与可扩展学习之间的平衡问题，实现紧凑的闭式表达和可扩展的原始发现机制。

摘要：

符号发现 governing equations 是科学机器学习中的长期目标，但解释性和可扩展学习之间仍存在根本性的权衡。经典符号回归方法产生显式解析表达式但依赖组合搜索，而神经网络在数据和维度上扩展效率高但产生不透明的表示。本文介绍 Symbolic Kolmogorov-Arnold Networks (Symbolic-KANs)，一种神经架构，通过在可训练深度网络中嵌入离散符号结构来弥合这一差距。Symbolic-KANs 将多变量函数表示为学习的单变量原语的组合，应用于学习的标量投影，受分析原语库、层次门控和符号正则化的指导，逐步将连续混合转化为一热选择。经过门控训练和离散化后，每个活跃单元选择一个单个原语和投影方向，产生紧凑的闭式表达式，而无需事后符号拟合。Symbolic-KANs 进一步充当可扩展的原始发现机制，识别最相关的分析组件，这些组件可以后续用于稀疏方程学习方法的候选库。我们证明 Symbolic-KAN 可靠地在数据驱动回归和逆动态系统中恢复正确的原语项和 governing 结构。此外，该框架扩展到偏微分方程的正向和逆物理启发学习，从 governing 约束中直接生成准确的解，同时构建紧凑的符号表示，其选择的原语反映底层方程的真实分析结构。这些结果使 Symbolic-KAN 成为可扩展、可解释和机理基础学习 governing 法律的一步。

英文摘要：

Symbolic discovery of governing equations is a long-standing goal in scientific machine learning, yet a fundamental trade-off persists between interpretability and scalable learning. 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. In this work, we introduce Symbolic Kolmogorov-Arnold Networks (Symbolic-KANs), a neural architecture that bridges this gap by embedding discrete symbolic structure directly within a trainable deep network. Symbolic-KANs represent multivariate functions as compositions of learned univariate primitives applied to learned scalar projections, guided by a library of analytic primitives, hierarchical gating, and symbolic regularization that progressively sharpens continuous mixtures into one-hot selections. After gated training and discretization, each active unit selects a single primitive and projection direction, yielding compact closed-form expressions without post-hoc symbolic fitting. Symbolic-KANs further act as scalable primitive discovery mechanisms, identifying the most relevant analytic components that can subsequently inform candidate libraries for sparse equation-learning methods. We demonstrate that Symbolic-KAN reliably recovers correct primitive terms and governing structures in data-driven regression and inverse dynamical systems. Moreover, the framework extends to forward and inverse physics-informed learning of partial differential equations, producing accurate solutions directly from governing constraints while constructing compact symbolic representations whose selected primitives reflect the true analytical structure of the underlying equations. These results position Symbolic-KAN as a step toward scalable, interpretable, and mechanistically grounded learning of governing laws.

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