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

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.

Figures (6)

• Figure 1: A schematic layout of the operations performed by a unit within a layer in the proposed Symbolic--KAN framework.
• Figure 2: Symbolic--KAN reconstruction result for the target function $F(x)=x^2$ in the data-driven regression experiment (Section \ref{['Sec:datadriven']}). The upper-left panels illustrate how the model learns both the function's symbolic structure and its numerical behavior. Through gated optimization, only the relevant primitive corresponding to the linear and quadratic terms are retained, while redundant components are suppressed, yielding a compact interpretable form consistent with the ground truth. The final prediction closely matches the target function across the training region and extends smoothly beyond it, demonstrating accurate interpolation and, to some extent, extrapolation performance with consistently low residual error.
• Figure 3: Comparison between the ground truth and Symbolic--KAN predicted solutions for the Van der Pol problem (Section \ref{['Sec:vanderpol']}) over two observation horizons, $T=20$ (top row) and $T=50$ (bottom row). All experiments are conducted with $|\mathcal{S}_r| = 10{,}000$, $N_{\mathrm{tr}}=100$, and the network configuration $[L,K_{\ell},E]=[4,6,3]$.
• Figure 4: Performance assessment of the proposed Symbolic--KAN method for the Van der Pol problem (Section \ref{['Sec:vanderpol']}). The relative errors of the reconstructed solution and the identified parameters $a$, $\mu$, and $c$ are examined under systematic variations of the data and architectural hyperparameters. The figure illustrates the effect of changing the number of training samples $N_{\mathrm{tr}}$ while keeping the number of interior collocation points and the network configuration fixed; varying the number of interior collocation points $|\mathcal{S}_r|$ for a fixed training set and architecture; modifying the network depth $L$ while maintaining the remaining hyperparameters fixed; and adjusting the number of neurons per layer, $K_{\ell}$, with all other settings unchanged. The results highlight the robustness and stability of the proposed framework with respect to both data availability and network design choices.
• Figure 5: Comparative evaluation of Symbolic--KAN, cPIKAN, and PINN for the 1D reaction--diffusion problem (Section \ref{['Exam.RD']}) on the extended domain $x \in [-4,4]$. The predicted reaction coefficient $\hat{\kappa}$ is also reported in each case. All network configurations and training settings are consistent with those listed in Table \ref{['tab:RD-symbolic']}. The comparison highlights the accuracy of the proposed Symbolic--KAN framework in both solution reconstruction and parameter identification relative to vanilla cPIKAN and PINN.