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.

翻译：控制方程的符号发现是科学机器学习中一个长期追求的目标，但可解释性与可扩展学习之间始终存在根本性权衡。经典符号回归方法能生成显式解析表达式，却依赖于组合搜索；而神经网络虽能高效处理数据和维度，但产生不透明的表征。在本文中，我们提出符号柯尔莫戈洛夫—阿诺德网络（Symbolic-KAN），这是一种通过将离散符号结构直接嵌入可训练深度网络来弥合这一差距的神经架构。Symbolic-KAN将多元函数表示为学习到的标量投影上应用的、学习到的单变量原语成分的组合，该过程由解析原语库、层次化门控和符号正则化引导，逐步将连续混合锐化为独热选择。经过门控训练和离散化后，每个活跃单元选择单一原语和投影方向，从而在不需事后符号拟合的情况下生成紧凑型闭式表达式。Symbolic-KAN还作为可扩展的原语发现机制，识别最相关的解析成分，这些成分随后可为稀疏方程学习方法提供候选库。我们证明，Symbolic-KAN能在数据驱动回归和逆动力学系统中可靠地恢复正确的原语项和控制结构。此外，该框架可扩展至偏微分方程的正向和逆向物理信息学习，直接从控制约束生成精确解，同时构建紧凑符号表示，其选定原语反映了底层方程的真实解析结构。这些结果将Symbolic-KAN定位为迈向可扩展、可解释且基于机制的定律学习的一步。