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

Salah A Faroughi; Farinaz Mostajeran; Amirhossein Arzani; Shirko Faroughi

cs.LGMar 25, 2026

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

Authors: Salah A Faroughi, Farinaz Mostajeran, Amirhossein Arzani, Shirko Faroughi

Organizations: Energy & Intelligence Lab, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA · Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA · Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, Utah, 84112, USA · Department of Biomedical Engineering, University of Utah, Salt Lake City, Utah, 84112, USA · Department of Mechanical Engineering, School of Engineering, Urmia University of Technology, Urmia, Iran

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 & tables

Figure 1: A schematic layout of the operations performed by a unit within a layer in the proposed Symbolic–KAN framework.

Target Function	$L,K_{\ell},E$	$\mathcal{E}(F_{\bm{\theta}})$	Selected Primitives, Edge & Unit
$F(x)=x^{2}$ $x\in[0,5]$ $N_{tr}$ = 250	2,2,3	$1.04\times 10^{-5}$	$\ell=0,k=0$ : \| $x,x^{2},x$ \| $\to$ [ $x$ ] (alive) $\ell=0,k=1$ : \| $x^{2}$ , cos, sin\| $\to$ [sin] (killed) $\ell=1,k=0$ : \| $x^{2},x^{2},x$ \| $\to$ [ $x^{2}$ ] (killed) $\ell=1,k=1$ : \| $x^{2}$ , exp, cos\| $\to$ [ $x^{2}$ ] (alive)
$F(x)=\displaystyle{\frac{\sin(3x)}{1+x^{2}}+0.4\cos(5x)}$ $x\in[0,5]$ $N_{tr}$ = 650	2,3,3	$7.75\times 10^{-3}$	$\ell=0,k=0$ : \|cos, $x$ , $x^{2}$ \| $\to$ [ $x^{2}$ ] (alive) $\ell=0,k=1$ : \| $x$ , exp, sin\| $\to$ [ sin ] (alive) $\ell=1,k=0$ : \|lorentz, sin, $x$ \| $\to$ [ sin ] (alive) $\ell=1,k=1$ : \| $x^{2}$ , $x^{2}$ , sin\| $\to$ [ sin ] (alive) $\ell=2,k=0$ : \|1, cos, exp\| $\to$ [ cos ] (alive) $\ell=2,k=1$ : \|sin, cos, lorentz\| $\to$ [ lorentz ] (alive)

Table 1: Results of the data-driven regression tests for 1D functions (Section 3.1 ). For each target function, we report the network configuration

(L,K_{\ell},E)

, the number of training samples

N_{\mathrm{tr}}

, the relative prediction error

\mathcal{E}(F_{\bm{\theta}})

, and the discrete symbolic primitives selected by Symbolic–KAN before (reported in

|\cdot|

) and after the hardening stage, indicated by

\to

symbol. Entries marked as alive denote the unit retained in the final symbolic structure, whereas killed indicates units pruned during training.

Figure 2: Symbolic–KAN reconstruction result for the target function $F(x)=x^{2}$ in the data-driven regression experiment (Section 3.1 ). 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.

Domain	$\mathcal{E}(u_{\bm{\theta}})$	$a$	$\mu$	$c$	Selected Primitives
$t\in[0,20]$	$6.02\times 10^{-4}$	$1.0000$	$0.0099$	$0.9998$	Layer 0: $\to$ $[\sin,\sin,\sin,\sin,\sin,x]$ Layer 1: $\to$ $[x,\cos,x,\cos,\cos,\sin]$ Layer 2: $\to$ $[x,x,\sin,\sin,\cos,\cos]$ Layer 3: $\to$ $[\sin,\cos,\sin,\sin,x,\cos]$
$t\in[0,50]$	$5.87\times 10^{-3}$	$0.9999$	$0.0093$	$0.9992$	Layer 0: $\to$ $[\cos,\cos,\cos,\sin,\cos,\sin]$ Layer 1: $\to$ $[\sin,\sin,\cos,x,\sin,\sin]$ Layer 2: $\to$ $[\sin,\sin,\cos,\sin,\cos,x]$ Layer 3: $\to$ $[\cos,\sin,x,\cos,\cos,\sin]$

Table 2: Results of the Van der Pol problem (Section 3.2.1 ). All experiments use

|\mathcal{S}_{r}|=10{,}000

and

N_{\mathrm{tr}}=100

. The validation error

\mathcal{E}(u_{\bm{\theta}})

, representing the mean relative

L^{2}

error averaged over the state variables is reported along with the predicted parameters

a

\mu

, and

c

. In this experiment, we employ the function library

\{\sin,\cos,\exp,x,x^{2}\}

and the network configuration

[L,K_{\ell},E]=[4,6,3]

. The resulting symbolic primitives chosen after hardening are reported for each unit in each layer. Note that, in this example, the unit gates were disabled (i.e., no unit pruning).

Figure 3: Comparison between the ground truth and Symbolic–KAN predicted solutions for the Van der Pol problem (Section 3.2.1 ) 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 3.2.1 ). 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.

Domain	Method	$L,K_{\ell},E$	$\mathcal{E}(u_{\bm{\theta}})$	$\kappa$	Selected Primitives
$x\in[-2,2]$	Symbolic–KAN	$4,6,3$	$5.93\times 10^{-4}$	$0.6994$	Layer 0: $\to$ $[\cos,\exp,\exp,x^{2},x^{2},x^{2}]$ Layer 1: $\to$ $[\cos,x,\sin,\sin,\sin,x^{2}]$ Layer 2: $\to$ $[\sin,x,\sin,\sin,x,\cos]$ Layer 3: $\to$ $[\cos,\cos,x^{2},\cos,x,\cos]$
$x\in[-4,4]$	Symbolic–KAN	$4,6,3$	$9.37\times 10^{-3}$	$0.6985$	Layer 0: $\to$ $[\sin,x,\cos,\sin,\sin,x]$ Layer 1: $\to$ $[\sin,\sin,\sin,x^{2},\sin,\sin]$ Layer 2: $\to$ $[\cos,\sin,\cos,\cos,\sin,\sin]$ Layer 3: $\to$ $[\sin,\cos,\sin,\cos,\sin,\sin]$
Comparison with baseline models (no symbolic discovery)
Domain	Method	$L,K_{\ell},k_{\mathrm{Cheb}}$	$\mathcal{E}(u_{\bm{\theta}})$	$\kappa$
$x\in[-2,2]$	cPIKAN	$4,6,3$	$8.25\times 10^{-4}$	$0.6998$	–
	PINN	$4,12,-$	$1.50\times 10^{-1}$	$0.6899$	–

Table 3: Results for the 1D reaction–diffusion problem (Section 3.3.1 ). All experiments are conducted with

|\mathcal{S}_{r}|=5{,}000

interior collocation points and

N_{\mathrm{tr}}=100

training samples. The table reports, for each method, the network configuration

(L,K_{\ell},E)

, the validation error

\mathcal{E}(u_{\bm{\theta}})

, and the identified reaction coefficient

\kappa

. For the Symbolic–KAN model, the selected symbolic primitives after the hardening stage are also listed. The candidate library used for symbolic discovery is

\{\sin,\cos,\exp,x,x^{2}\}

. For baseline models,

k_{\mathrm{Cheb}}

denotes the degree of the Chebyshev polynomial expansion used in cPIKAN. Note that, in this example, the unit gates were disabled (i.e., no unit pruning).

Figure 5: Comparative evaluation of Symbolic–KAN, cPIKAN, and PINN for the 1D reaction–diffusion problem (Section 3.3.1 ) 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 3 . The comparison highlights the accuracy of the proposed Symbolic–KAN framework in both solution reconstruction and parameter identification relative to vanilla cPIKAN and PINN.

Method	$L,K_{\ell},E$	$\mathcal{E}(u_{\bm{\theta}})$	Selected Primitives
Symbolic–KAN	$4,6,3$	$1.11\times 10^{-3}$	Layer 0: $\to$ $[\cos,\sin,\cosh,\sinh,\cosh,x^{2}]$ Layer 1: $\to$ $[\sin,x,x,\sin,x,\sin]$ Layer 2: $\to$ $[\text{const},\sin,\sin,\sinh,\cos,\cos]$ Layer 3: $\to$ $[\sinh,\cosh,\sinh,\sin,\cosh,\cosh]$
Comparison with baseline models (no symbolic discovery)
Method	$L,K_{\ell},k_{\mathrm{Cheb}}$	$\mathcal{E}(u_{\bm{\theta}})$
cPIKAN	$4,6,3$	$8.76\times 10^{-3}$	–
PINN	$4,18,-$	$2.71\times 10^{-3}$	–

Table 4: Results for the Laplace problem (Section 3.3.2 ). All experiments are conducted with

|\mathcal{S}_{r}|=10{,}000

interior collocation points and

|\mathcal{S}_{b}|=400

boundary collocation points. For each method, the network configuration

(L,K_{\ell},E)

, and the validation error

\mathcal{E}(u_{\bm{\theta}})

are reported. For the Symbolic–KAN model, the selected symbolic primitives after the hardening stage are also listed. The candidate library used for symbolic discovery is

\{1,x,x^{2},\sin,\cos,\sinh,\cosh,\exp\}

. For baseline models,

k_{\mathrm{Cheb}}

denotes the degree of the Chebyshev polynomial expansion used in cPIKAN. Note that, in this example, the unit gates were disabled (i.e., no unit pruning).

Figure 6: Comparative evaluation of Symbolic–KAN, cPIKAN, and PINN for Laplace equation (Section 3.3.2 ). All network configurations and training settings are listed in Table 4 . The comparison highlights the accuracy of the proposed Symbolic–KAN framework relative to vanilla cPIKAN and PINN.

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Abstract

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