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

2026-03-24

AI Summary

This work addresses the fundamental trade-off in scientific machine learning between interpretability and scalability: symbolic regression struggles to scale, while neural networks lack explicit analytical expressions. To bridge this gap, the authors propose Symbolic-KAN, which for the first time embeds discrete symbolic structures into Kolmogorov–Arnold Networks. By integrating learnable univariate analytic primitives, scalar projections, hierarchical gating, and symbolic regularization, the model enables end-to-end training and automatically sharpens into compact symbolic expressions without post-processing. The framework supports automatic discovery of analytic primitives and accurately recovers both the structure and constituent functions of ground-truth governing equations across diverse tasks—including data-driven regression, inverse dynamical systems, and physics-informed partial differential equations—achieving high accuracy while maintaining strong interpretability.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

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.

Problem

• symbolic discovery
• governing equations
• interpretability
• scalable learning
• scientific machine learning

Innovation

• Symbolic-KAN
• interpretable learning
• symbolic regression
• Kolmogorov-Arnold Networks
• physics-informed learning