Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning
arXiv:2603.23854v1 Announce Type: new Abstract: 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
arXiv:2603.23854v1 Announce Type: new Abstract: 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.
Executive Summary
This article introduces Symbolic Kolmogorov-Arnold Networks (Symbolic-KANs), a novel neural architecture that bridges the gap between interpretability and scalable learning in scientific machine learning. By embedding discrete symbolic structure within a trainable deep network, Symbolic-KANs represent multivariate functions as compositions of learned univariate primitives. This approach enables the identification of relevant analytic components, facilitating the construction of compact symbolic representations. The framework is demonstrated to be effective in data-driven regression, inverse dynamical systems, and physics-informed learning of partial differential equations. The results position Symbolic-KAN as a step toward scalable, interpretable, and mechanistically grounded learning of governing laws.
Key Points
- ▸ Symbolic-KANs bridge the gap between interpretability and scalable learning in scientific machine learning.
- ▸ The architecture represents multivariate functions as compositions of learned univariate primitives.
- ▸ Symbolic-KANs are demonstrated to be effective in various applications, including data-driven regression and physics-informed learning.
Merits
Strength in Interpretability
Symbolic-KANs provide an interpretable representation of complex functions, enabling the identification of relevant analytic components and the construction of compact symbolic expressions.
Scalability
The architecture scales efficiently with data and dimensionality, making it suitable for large-scale applications.
Mechanistically Grounded Learning
Symbolic-KANs learn governing laws directly from data, reflecting the true analytical structure of the underlying equations.
Demerits
Computational Complexity
The training process of Symbolic-KANs may be computationally demanding due to the hierarchical gating and symbolic regularization mechanisms.
Limited Generalizability
The framework may require significant modifications to accommodate different problem domains and applications.
Expert Commentary
The introduction of Symbolic-KANs represents a significant advancement in the field of scientific machine learning. By bridging the gap between interpretability and scalability, the framework enables the development of accurate and interpretable models in various applications. The results demonstrate the effectiveness of Symbolic-KANs in data-driven regression, inverse dynamical systems, and physics-informed learning of partial differential equations. However, the computational complexity and limited generalizability of the framework are notable limitations that require further investigation. Nevertheless, the potential of Symbolic-KANs to revolutionize the field of scientific machine learning is substantial, and the framework is likely to have significant implications for various applications and policy-making decisions.
Recommendations
- ✓ Further investigation into the computational complexity and scalability of Symbolic-KANs is necessary to ensure their applicability in large-scale applications.
- ✓ The framework should be extended to accommodate different problem domains and applications, requiring modifications to the architecture and training process.
Sources
Original: arXiv - cs.LG
