Softly Symbolifying Kolmogorov-Arnold Networks

James Bagrow; Josh Bongard

cs.LGNov 27, 2025

Softly Symbolifying Kolmogorov-Arnold Networks

Authors: James Bagrow, Josh Bongard

Organizations: Mathematics & Statistics, University of Vermont, Burlington, VT, United States · Vermont Complex Systems Center, University of Vermont, Burlington, VT, United States · Computer Science, University of Vermont, Burlington, VT, United States

Abstract

Kolmogorov--Arnold Networks (KANs) offer a promising path toward interpretable machine learning: their learnable activations can be studied individually while collectively fitting complex data accurately. In practice, however, trained activations often lack symbolic fidelity, learning pathological decompositions with no meaningful correspondence to interpretable forms. We propose Softly Symbolified Kolmogorov--Arnold Networks (S2KAN), which integrate symbolic primitives directly into training. Each activation draws from a dictionary of symbolic and dense terms, with learnable gates that sparsify the representation. Crucially, this sparsification is differentiable, enabling end-to-end optimization, and uses an optimization objective inspired by minimum description length. S2KAN can discover interpretable activation functions while retaining dense splines for flexible approximation. We introduce a quantitative measure of symbolic fidelity, assessing how closely learned activations can be approximated by short expressions from a specified symbolic language. Across real-world prediction tasks, dynamical-system forecasting, and synthetic symbolic benchmarks, S2KAN generally achieves higher symbolic fidelity and often far smaller models, while maintaining competitive predictive accuracy on real-world data and low prediction errors on most symbolic benchmark configurations. We also examine the models' learned responses on scientific data, including the contributions of their symbolic components. Together, these results show that incorporating symbolic structure during training can yield compact, accurate KAN models whose learned responses can be examined through their symbolic components.

Figures & tables

Figure 1: Learning

y=\mathrm{sinc}(x)=\sin(x)/x

with a multiplicative Kolmogorov–Arnold network (KAN). The standard KAN lacks symbolic fidelity, exhibiting a numerically accurate but otherwise pathological decomposition, whereas S2KAN captures the underlying sinc decomposition. The two models have empirical symbolic fidelities of 0% and 100%, respectively; see Methods (App. A ) for evaluation details.

Airfoil : shape $[5,5,1]$ ; RMSE in dB
	Test accuracy		Model size		SF (%), $\epsilon=0.01$			SF (%), $\epsilon=0.1$
Model	$R^{2}$	RMSE	Active	$k$	$J=1$	$J=2$	$J=3$	$J=1$	$J=2$	$J=3$
KAN	0.911 [-1pt][0.902, 0.918]	2.05 [-1pt][1.96, 2.13]	30 [-1pt] [0.0, 0.0]	450 [-1pt] [0.0, 0.0]	10.5 [-1pt][7.2, 13.5]	26.8 [-1pt][24.4, 28.6]	28.6 [-1pt][28.6, 31.0]	14.3 [-1pt][11.1, 17.7]	39.3 [-1pt][36.0, 46.0]	59.6 [-1pt][51.8, 62.1]
S2KAN	0.912 [-1pt][0.909, 0.920]	2.02 [-1pt][1.95, 2.05]	26 [-1pt][24.2, 26.0]	103.8 [-1pt][94.7, 114.1]	22.4 [-1pt][20.0, 25.7]	50.0 [-1pt][46.3, 52.2]	64.0 [-1pt][60.0, 69.2]	33.3 [-1pt][29.4, 39.8]	84.3 [-1pt][82.6, 87.5]	91.3 [-1pt][87.5, 92.0]
Concrete : shape $[13,32,1]$ ; RMSE in MPa
KAN	0.915 [-1pt][0.898, 0.925]	4.90 [-1pt][4.49, 5.23]	448 [-1pt] [0.0, 0.0]	6720 [-1pt] [0.0, 0.0]	0.0 [-1pt][0.0, 0.3]	2.6 [-1pt][1.9, 3.4]	12.9 [-1pt][11.6, 13.9]	26.2 [-1pt][19.3, 29.0]	61.5 [-1pt][47.2, 65.1]	83.2 [-1pt][71.5, 86.9]

Table 1: Prediction, model size, and symbolic fidelity on four real-world datasets. S2KAN yields substantially smaller models and higher symbolic fidelity, with modest differences in predictive accuracy. Test accuracy is reported as

R^{2}

and root mean squared error (RMSE). Values are medians over ten data splits with three initialization seeds per split; brackets are interquartile ranges (IQRs; omitted for constant values). S2KAN uses

\beta=1

after warmup. “Active” counts gated-on activation functions, and

k

denotes the effective parameter count

k_{w}

from Eq. ( 17 ). Symbolic fidelity (SF, percent; Section 3 ) is the fraction of functionally active edges approximated within relative tolerance

\epsilon

using at most

J

terms.

Figure 2: Illustrative model-response analysis of predicted concrete strength development with age. Consistent with expected curing behavior, the complete model predicts strength increasing with age at an overall slowing rate. The dominant branch exhibits a decreasing rate of gain throughout the evaluated interval. a , Four hidden branches for a fixed recipe, ranked by total variation of their output contributions over days 3–365. Each

b_{j}

sums the learned input activations evaluated at the recipe’s 11 fixed non-age features. Inset formulas show the learned functions, with branch-specific coefficients

c_{kj}

;

z_{\log(\mathrm{age}+1)}

denotes standardized log-age, and

\sqrt{h}_{*}

denotes the protected square root. These branches have no active raw-age or dense terms. Output curves are converted to MPa and centered at day 3. b , Branch contributions and complete S2KAN and KAN predictions relative to day 3. All 32 branches remain in the S2KAN total. Both models use the same data split and initialization seed; the recipe was selected by proximity to the median standardized ingredient composition among 181 eligible recipes, without using strengths or predictions. Here

s_{y}

is the population standard deviation of the fitting-set targets.

Figure 3: Learning dynamics for S2KAN on four real-world datasets, with

\beta=1

after 500 warmup epochs. Rows show Airfoil, Concrete, Power plant, and Superconductor. Columns show mean gate-opening probabilities by basis type; total gate entropy and decisiveness; and normalized per-layer entropy

H/H_{\max}

. Dense denotes the spline branch, including its SiLU residual. Curves show medians over ten data splits with three initialization seeds per split; bands show interquartile ranges. Gray shading marks warmup (

\beta=0

). Full 5,000-epoch trajectories are shown.

Figure 4: Expected fraction of active gates per layer,

\rho_{\ell}

(Eq. ( 21 )), for the real-world and dynamical-systems S2KAN models. Bars show medians; whiskers indicate IQRs across ten initialization seeds per dynamical-system configuration and 30 fits per real-world dataset (ten data splits and three initialization seeds per split). Overall fractions are computed within each model before aggregation.

Method	Gradient-based	Compositional	Symbolic	Graceful degradation
SINDy ( Brunton et al., (2016) )	✗	✗	✓	✗
PySR ( Cranmer, (2023) )	✗	✓	✓	✗
AI Feynman ( Udrescu & Tegmark, (2020) ; Udrescu et al., (2020) )	✗ a	✓	✓	✗
EQL / iEQL ( Sahoo et al., (2018) ; Werner et al., (2021) )	✓	✓	✓	✗
KAN ( Liu, Wang et al., (2025) )	✓	✓	✗ b	n/a
Projective KAN ( Poole et al., (2026) )	✓	✓	✗ b	n/a

Table 4: Qualitative comparison of S2KAN with selected symbolic regression and KAN methods.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Ikeda, $[2,4,4,4,2]$ , $\beta=0$
	Edges				Open gates
Layer	Edges	Active	$\mathcal{S}$ -only	$\mathcal{S}{+}\mathcal{F}$	$\mathcal{S}$	$\mathcal{F}$	$\mathcal{R}$
L0 ( $2\!\to\!4$ )	8 [-1pt] [0, 0]	8 [-1pt] [0, 0]	0 [-1pt] [0, 0]	0 [-1pt][0, 0]	0 [-1pt] [0, 0]	12.5 [-1pt][11.25, 13]	8 [-1pt][8, 8]
L1 ( $4\!\to\!4$ )	16 [-1pt] [0, 0]	15.5 [-1pt][15, 16]	0 [-1pt] [0, 0]	0 [-1pt][0, 0]	0.5 [-1pt][0, 1]	20.5 [-1pt][17.25, 21.75]	15 [-1pt][14.25, 16]
L2 ( $4\!\to\!4$ )	16 [-1pt] [0, 0]	14 [-1pt][13.25, 14.75]	0 [-1pt][0, 0]	3 [-1pt][2, 4.75]	2.5 [-1pt][1, 3.75]	24 [-1pt][22, 26.75]	10.5 [-1pt][9, 12]
L3 ( $4\!\to\!2$ )	8 [-1pt] [0, 0]	8 [-1pt][8, 8]	0 [-1pt][0, 0]	2 [-1pt][2, 2]	4 [-1pt][3.25, 4.75]	15.5 [-1pt][13, 16.75]	6 [-1pt][6, 6]

Appendix

Table 5: Sparsity breakdown for the dynamical-systems S2KAN models. Entries are medians over ten initialization seeds, with first and third quartiles below (omitted for constant counts). Ikeda uses the direct map at 20,000 epochs with

P=Q=4

; Ecosystem uses the residual map at 45,000 epochs with

P=0

Q=4

. Spline grids are updated early in training. Active means at least one open gate;

\mathcal{S}

-only excludes both

\mathcal{F}

and

\mathcal{R}

, while

\mathcal{S}{+}\mathcal{F}

excludes

\mathcal{R}

. Gates are counted at

p_{\mathrm{open}}>0.5

\mathcal{S}

is the symbolic library,

\mathcal{F}

is Chebyshev/Fourier, and

\mathcal{R}

is the spline branch (one gate per edge, not one per spline coefficient). Ecosystem's three fixed identity additions are excluded. Totals are computed within each model before aggregation, so their medians need not equal sums of layer medians.

Airfoil, $[5,5,1]$ , $\beta=1$
	Edges				Open gates
Layer	Edges	Active	$\mathcal{S}$ -only	$\mathcal{S}{+}\mathcal{F}$	$\mathcal{S}$	$\mathcal{F}$	$\mathcal{R}$
L0 ( $5\!\to\!5$ )	25 [-1pt] [0, 0]	21 [-1pt][19.25, 21]	6 [-1pt][5, 7]	18 [-1pt][17.25, 19]	27 [-1pt][25, 31]	20 [-1pt][17, 21]	2 [-1pt][2, 2]
L1 ( $5\!\to\!1$ )	5 [-1pt] [0, 0]	5 [-1pt] [0, 0]	0 [-1pt] [0, 0]	4 [-1pt][4, 5]	7 [-1pt][5, 7.75]	10 [-1pt][9, 11]	1 [-1pt][0, 1]
Total	30 [-1pt] [0, 0]	26 [-1pt][24.25, 26]	6 [-1pt][5, 7]	23 [-1pt][22, 23.75]	33.5 [-1pt][31, 36]	30.5 [-1pt][27, 32]	3 [-1pt][2, 3]
Concrete, $[13,32,1]$ , $\beta=1$

Appendix

Table 6: Sparsity breakdown for the real-world S2KAN models. All use

\beta=1

P=0

Q=2

. Entries are medians over ten data splits with three initialization seeds per split (30 fits per dataset); brackets below give the first and third quartiles, omitted for constant counts. These are descriptive intervals. Gate definitions and within-model totals follow the dynamical-systems table (Table 5 ).

Airfoil
			Inference
Model	Training (min)		Batch 1 ( $\mu\mathrm{s}$ )		Batch 256 ( $\mu\mathrm{s}$ )
KAN	2.77	[2.73, 2.80]	452.7	[442.0, 472.1]	1128.0	[1106.5, 1172.0]
S2KAN	4.34	[4.24, 4.42]	514.0	[318.4, 531.0]	731.6	[558.3, 842.8]
Concrete
KAN	4.71	[3.93, 4.94]	484.7	[473.0, 490.2]	3427.9	[3352.8, 3806.8]

Appendix

Table 7: Training and inference wall times for the models in Table 1 , on an AMD EPYC 9654 (single-threaded). Entries are medians [first quartile, third quartile] over 30 fits per row. S2KAN takes about 1.6 times as long to train but generally provides faster inference.

Model	$\beta$	Defined	$\lambda_{1}$	$\lambda_{2}$	$\lambda_{3}$	$\|\Delta\lambda_{1}\|$
Ikeda map : exponents per iterate
Reference	—	—	0.508	-0.719	—	—
KAN	—	10/10 [-1pt] [0.0, 0.0]	0.508 [-1pt][0.505, 0.511]	-0.727 [-1pt][-0.731, -0.722]	— [-1pt] [0.0, 0.0]	0.0059 [-1pt][0.0040, 0.0094]
S2KAN	0	10/10 [-1pt] [0.0, 0.0]	0.508 [-1pt][0.506, 0.512]	-0.725 [-1pt][-0.731, -0.723]	— [-1pt] [0.0, 0.0]	0.0074 [-1pt][0.0046, 0.0108]
S2KAN	1	10/10 [-1pt] [0.0, 0.0]	0.508 [-1pt][0.501, 0.512]	-0.727 [-1pt][-0.729, -0.725]	— [-1pt] [0.0, 0.0]	0.0068 [-1pt][0.0037, 0.0143]
S2KAN	3	10/10 [-1pt] [0.0, 0.0]	0.507 [-1pt][0.502, 0.508]	-0.742 [-1pt][-0.756, -0.730]	— [-1pt] [0.0, 0.0]	0.0044 [-1pt][0.0009, 0.0115]

Appendix

Table 8: Full Lyapunov spectra of the models in Table 2 . Entries are medians [first quartile, third quartile] over defined spectra; “Defined” gives their count out of ten fits. The Ikeda map has two exponents, so

\lambda_{3}

is inapplicable. Reference entries are finite-orbit estimates for the generating map.

|\Delta\lambda_{1}|

is the absolute difference between leading-exponent estimates from the first and second halves of the evaluated orbit.

Figure 5: Temporal structure of the learned dynamical systems measured using autocorrelation functions (left) and power spectral densities (right), for each state coordinate. Colored curves and shading show pointwise medians and IQRs across completed fits; black curves are reference block means. All shown Ikeda, KAN, and $\beta=0$ configurations contribute 10/10 fits. Ecosystem S2KAN at $\beta=1$ contributes nine PSDs and eight ACFs: one trajectory diverges and another is constant, making its ACF undefined. Other completed trajectories remain included.

Ikeda map
Model	$\beta$	Multi-step RMSE	Horizon	Late overlap
KAN	—	0.858 [-1pt][0.851, 0.871]	9.0 [-1pt][8.5, 9.0]	0.913 [-1pt][0.910, 0.914]
S2KAN	0	0.866 [-1pt][0.856, 0.876]	9.0 [-1pt][8.0, 10.0]	0.913 [-1pt][0.909, 0.914]
S2KAN	1	0.861 [-1pt][0.855, 0.871]	8.5 [-1pt][7.6, 9.0]	0.909 [-1pt][0.907, 0.912]
S2KAN	3	0.873 [-1pt][0.863, 0.875]	7.2 [-1pt][6.5, 7.5]	0.904 [-1pt][0.897, 0.910]
Ecosystem

Appendix

Table 9: Rollout diagnostics for the models in Table 2 . Entries are medians [first quartile, third quartile] over ten fits, with undefined multi-step RMSEs excluded as described in the text. Horizon is in discrete steps and late overlap lies in

[0,1]

Figure 6: Self-sparsification of the Ikeda map (a) and Ecosystem networks (b), both at

\beta=0

. Solid curves show medians across ten seeds; shading indicates the interquartile range. Horizontal dashed lines indicate the effective parameter count

k

of baseline KAN. Counts are expected effective parameter counts, rather than hard gate or active-edge counts. The first point follows the first completed epoch. Training MSEs are epoch-averaged stochastic minibatch values; shading shows descriptive interquartile ranges, not confidence intervals.

Airfoil : RMSE in dB
		SF (%), $\epsilon=0.01$			SF (%), $\epsilon=0.1$
KAN loss	RMSE	$J=1$	$J=2$	$J=3$	$J=1$	$J=2$	$J=3$
Raw MSE	2.047 [-1pt][1.963, 2.132]	10.5 [-1pt][7.2, 13.5]	26.8 [-1pt][24.4, 28.6]	28.6 [-1pt][28.6, 31.0]	14.3 [-1pt][11.1, 17.7]	39.3 [-1pt][36.0, 46.0]	59.6 [-1pt][51.8, 62.1]
Log MSE	2.021 [-1pt][1.972, 2.110]	11.1 [-1pt][10.3, 14.2]	25.9 [-1pt][25.0, 28.6]	28.6 [-1pt][28.6, 31.0]	14.8 [-1pt][13.8, 17.9]	43.7 [-1pt][37.6, 44.8]	58.2 [-1pt][54.0, 65.2]
Concrete : RMSE in MPa
Raw MSE	4.900 [-1pt][4.488, 5.234]	0.0 [-1pt][0.0, 0.3]	2.6 [-1pt][1.9, 3.4]	12.9 [-1pt][11.6, 13.9]	26.2 [-1pt][19.3, 29.0]	61.5 [-1pt][47.2, 65.1]	83.2 [-1pt][71.5, 86.9]

Appendix

Table 11: Baseline KAN trained with raw or log MSE. Values are medians over thirty fits; smaller brackets give IQRs, omitted for constant values. All checkpoints are validation-selected. Parameter counts are unchanged by the loss choice.

Dataset	Model	RMSE	$R^{2}$	Active	$k$
Airfoil	Full S2KAN	2.022 [-1pt][1.954, 2.048]	0.912 [-1pt][0.909, 0.920]	26 [-1pt][24.2, 26.0]	103.8 [-1pt][94.7, 114.1]
	No-dense S2KAN	2.087 [-1pt][2.021, 2.148]	0.908 [-1pt][0.902, 0.915]	26 [-1pt][25.0, 26.0]	64.4 [-1pt][62.1, 69.8]
	Dense-only S2KAN	2.334 [-1pt][2.168, 2.414]	0.886 [-1pt][0.875, 0.898]	14 [-1pt][13.0, 16.0]	211.9 [-1pt][197.3, 240.0]
Concrete	Full S2KAN	5.268 [-1pt][4.996, 5.612]	0.894 [-1pt][0.885, 0.906]	61 [-1pt][58.0, 63.8]	76.2 [-1pt][71.7, 80.4]
	No-dense S2KAN	5.165 [-1pt][4.876, 5.404]	0.898 [-1pt][0.891, 0.909]	71 [-1pt][63.5, 75.8]	88.1 [-1pt][79.5, 94.6]
	Dense-only S2KAN	7.366 [-1pt][7.016, 7.878]	0.793 [-1pt][0.781, 0.809]	4 [-1pt][4.0, 5.0]	64.4 [-1pt][60.9, 71.4]

Appendix

Table 12: Activation-family ablations on four real-world datasets. Entries are medians over 30 fits per dataset and method, with IQRs in brackets (omitted for constant values).

			Model size		SF (%), $\epsilon=0.01$			SF (%), $\epsilon=0.1$
Problem	Model	RMSE	Active	$k$	$J=1$	$J=2$	$J=3$	$J=1$	$J=2$	$J=3$
F1	S2KAN	0.0055 [-1pt][0.0033, 0.0082]	6 [-1pt][6, 6]	42.5 [-1pt][38.4, 51.3]	40.0 [-1pt][21.3, 40.0]	80.0 [-1pt][76.2, 95.0]	100.0 [-1pt] [0.0, 0.0]	60.0 [-1pt][50.0, 60.0]	100.0 [-1pt][100.0, 100.0]	100.0 [-1pt] [0.0, 0.0]
	No-dense S2KAN	0.0041 [-1pt][0.0030, 0.0083]	6 [-1pt] [0.0, 0.0]	40.3 [-1pt][39.0, 42.2]	40.0 [-1pt][40.0, 47.5]	100.0 [-1pt][100.0, 100.0]	100.0 [-1pt] [0.0, 0.0]	60.0 [-1pt][60.0, 60.0]	100.0 [-1pt] [0.0, 0.0]	100.0 [-1pt] [0.0, 0.0]
F2	S2KAN	0.0063 [-1pt][0.0045, 0.0100]	6 [-1pt][6, 6]	41.4 [-1pt][40.0, 45.4]	60.0 [-1pt][52.5, 72.9]	100.0 [-1pt][80.0, 100.0]	100.0 [-1pt][100.0, 100.0]	100.0 [-1pt][76.2, 100.0]	100.0 [-1pt][100.0, 100.0]	100.0 [-1pt] [0.0, 0.0]
	No-dense S2KAN	0.0065 [-1pt][0.0035, 0.0076]	6 [-1pt] [0.0, 0.0]	41.3 [-1pt][38.6, 45.3]	55.0 [-1pt][40.0, 60.0]	100.0 [-1pt] [0.0, 0.0]	100.0 [-1pt] [0.0, 0.0]	90.0 [-1pt][80.0, 100.0]	100.0 [-1pt] [0.0, 0.0]	100.0 [-1pt] [0.0, 0.0]
F3	S2KAN	0.0062 [-1pt][0.0043, 0.0120]	6 [-1pt][6, 6]	55.9 [-1pt][53.2, 59.5]	40.0 [-1pt][28.7, 40.0]	80.0 [-1pt][75.0, 80.0]	80.0 [-1pt][80.0, 95.8]	60.0 [-1pt][42.5, 60.0]	100.0 [-1pt][85.0, 100.0]	100.0 [-1pt][100.0, 100.0]

Appendix

Table 13: Dense-branch ablation on Nguyen benchmarks, using architecture L and

\beta=1

for both methods. Entries are medians over ten initialization seeds on the same training/test sample; brackets give the first and third quartiles (omitted for constant values). “No-dense S2KAN” disables both spline and SiLU contributions while retaining the symbolic, Chebyshev and Fourier terms. RMSE is test error; Active counts gate-active edges;

k

is weighted expected support. SF is the percentage of functionally active edges approximated within relative RMSE tolerance

\epsilon

by at most

J

gauged terms from

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

. All fits are included; none has an empty functional-activity set.

Figure 7: Regularization trades model size against predictive accuracy on real-world datasets. Curves and shading show medians and IQRs over 30 validation-selected fits at each

\beta

. RMSE is expressed as a percentage of the test-target standard deviation (SD), calculated separately for each fit before aggregation. The dotted line marks

\beta=1

Figure 8: The same real-world regularization sweep shown in Fig. 7 but for fits evaluated at the end of training. Unlike the validation-selected fits, median size decreases throughout the sweep; excessive regularization produces small but inaccurate models.

Figure 9: Accuracy, model size, and stopping epoch versus regularization strength on Nguyen benchmarks. Each curve shows one problem’s median over ten initialization seeds; IQRs are omitted for readability. Test RMSE is expressed as a percentage of the test-target standard deviation, calculated per fit before aggregation;

k

follows Table 3 . The bottom row shows median stopping epochs under the gate-decisiveness rule, including runs reaching the 10,000-epoch cap (dashed line). The dotted line marks

\beta=1

. Positive

\beta

values are log spaced, with zero shown separately; the RMSE axis is logarithmic and the other axes linear.

Cohort	Paired / total	Equal scores	Largest $\|\Delta\|$	Largest median shift
Dynamical systems	80/80	440/480	5.88	2.27
Nguyen	600/600	3472/3600	50.00	28.57
No-dense control	100/100	547/600	50.00	27.50
Real world	240/240	1333/1440	33.33	17.88
Sinc illustration	2/2	12/12	0.00	0.00

Appendix

Table 14: Amplitude-constraint sensitivity of SF. Changes are constrained minus unconstrained SF, in percentage points. “Equal scores” counts identical values across all six settings for the paired models; settings are correlated. “Largest median shift” compares medians within the same dataset/problem, architecture, method and regularization group at each setting, matching the main tables’ aggregation. All expected fits are included in each group. The sinc illustration contains one checkpoint per method.

Comparison	Unconstrained	$C=3$
Nguyen	114/66/0	113/67/0
Real world	24/0/0	24/0/0
Dynamical systems	28/6/2	28/6/2
Full vs no dense	11/39/10	12/36/12

Appendix

Table 15: Method comparisons using median SF at each of the six settings. Entries are wins/ties/losses for S2KAN versus baseline KAN, except the final row, which compares full S2KAN with its no-dense control. Nguyen comparisons span ten problems and three architectures; real-world comparisons span four datasets; dynamical-system comparisons span six S2KAN configurations, each compared with its system’s baseline. Each median uses all fits in the corresponding paper table.

Figure 10: Unconstrained versus amplitude-constrained (

C=3

) symbolic fidelity for the 1,022 models in the main-results and Nguyen no-dense inventory, including both sinc illustrations. Each panel contains one point per model; coincident points overlap. The line is equality. The

J=1

search is unchanged by construction. Occasional points above the line reflect differences in the nonlinear search path rather than an improvement in the globally optimal constrained approximation.

Kolmogorov-Arnold Networks

Kolmogorov-Arnold Networks (KANs) learn compositional functions through trainable one-dimensional edge transformations for approximation and interpretable modeling.

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Interpretable ML Scientific ML Kolmogorov-Arnold Networks Symbolic Regression PDE Discovery Physics-Informed ML

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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.

Abstract

Kolmogorov-Arnold Networks

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