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

arXiv: https://arxiv.org/abs/2603.23854 · HTML: https://mlsanity.com/paper/2603.23854/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 69 references that are in the map.
- First posted on arXiv 25 Mar 2026.

## Builds on (most cited first, 30 of 40)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [Wav-KAN: Wavelet Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2405.12832/) (2024)
- [KAN 2.0: Kolmogorov-Arnold Networks Meet Science](https://mlsanity.com/paper/2408.10205/) (2024)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems](https://mlsanity.com/paper/2405.19143/) (2024)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [KAN or MLP: A Fairer Comparison](https://mlsanity.com/paper/2407.16674/) (2024)
- [KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics](https://mlsanity.com/paper/2407.04192/) (2024)
- [Post-Hoc Explanations Fail to Achieve their Purpose in Adversarial Contexts](https://mlsanity.com/paper/2201.10295/) (2022)
- [Kolmogorov-Arnold Network for Satellite Image Classification in Remote Sensing](https://mlsanity.com/paper/2406.00600/) (2024)
- [BSRBF-KAN: A combination of B-splines and Radial Basis Functions in Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2406.11173/) (2024)
- [Unveiling the Power of Wavelets: A Wavelet-based Kolmogorov-Arnold Network for Hyperspectral Image Classification](https://mlsanity.com/paper/2406.07869/) (2024)
- [ExSpliNet: An interpretable and expressive spline-based neural network](https://mlsanity.com/paper/2205.01510/) (2022)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theorem](https://mlsanity.com/paper/2308.05509/) (2023)
- [KKANs: Kurkova-Kolmogorov-Arnold Networks and Their Learning Dynamics](https://mlsanity.com/paper/2412.16738/) (2024)
- [End-to-end learnable EEG channel selection for deep neural networks with Gumbel-softmax](https://mlsanity.com/paper/2102.09050/) (2021)
- [SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2411.06286/) (2024)
- [ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification](https://mlsanity.com/paper/2410.16528/) (2024)
- [SCKansformer: Fine-Grained Classification of Bone Marrow Cells via Kansformer Backbone and Hierarchical Attention Mechanisms](https://mlsanity.com/paper/2406.09931/) (2024)
- [Representation Meets Optimization: Training PINNs and PIKANs for Gray-Box Discovery in Systems Pharmacology](https://mlsanity.com/paper/2504.07379/) (2025)
- [Explaining Deep Neural Networks](https://mlsanity.com/paper/2010.01496/) (2020)
- [Interpreting and generalizing deep learning in physics-based problems with functional linear models](https://mlsanity.com/paper/2307.04569/) (2023)
- [EPi-cKANs: Elasto-Plasticity Informed Kolmogorov-Arnold Networks Using Chebyshev Polynomials](https://mlsanity.com/paper/2410.10897/) (2024)

## Built on by (most cited first, 4 of 4)

- [Tackling Failure Modes of PINNs and PIKANs Using Conflict-Free Gradients](https://mlsanity.com/paper/2609.14841/) (2026)
- [Neural Discovery of Memory and Nonlocal Kernels in Integro-Differential Equations with Constrained Kolmogorov--Arnold Networks](https://mlsanity.com/paper/2607.11110/) (2026)
- [Hardware-Efficient Neuro-Symbolic Networks with the Exp-Minus-Log Operator](https://mlsanity.com/paper/2604.13871/) (2026)
- [Low-Rank Kolmogorov-Arnold Generator Flows for Interpretable Operator Learning](https://mlsanity.com/paper/2610.13028/) (2026)

## Abstract (arXiv)

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.