cs.LGMar 30, 2026

FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks

Authors: Gnankan Landry Regis N'guessan

Organizations: Axiom Research Group · Dept. of Applied Mathematics and Computational Science, NM-AIST, Arusha, Tanzania · AIMS Research and Innovation Centre, Kigali, Rwanda

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Abstract

Kolmogorov-Arnold Networks (KAN) employ B-spline bases on a fixed grid, providing no intrinsic multi-scale decomposition for non-smooth function approximation. We introduce Fractal Interpolation KAN (FI-KAN), which incorporates learnable fractal interpolation function (FIF) bases from iterated function system (IFS) theory into KAN. Two variants are presented: Pure FI-KAN (Barnsley, 1986) replaces B-splines entirely with FIF bases; Hybrid FI-KAN (Navascues, 2005) retains the B-spline path and adds a learnable fractal correction. The IFS contraction parameters give each edge a differentiable fractal dimension that adapts to target regularity during training. On a Holder regularity benchmark (α∈[0.2,2.0]α\in [0.2, 2.0]), Hybrid FI-KAN outperforms KAN at every regularity level (1.3x to 33x). On fractal targets, FI-KAN achieves up to 6.3x MSE reduction over KAN, maintaining 4.7x advantage at 5 dB SNR. On non-smooth PDE solutions (scikit-fem), Hybrid FI-KAN achieves up to 79x improvement on rough-coefficient diffusion and 3.5x on L-shaped domain corner singularities. Pure FI-KAN's complementary behavior, dominating on rough targets while underperforming on smooth ones, provides controlled evidence that basis geometry must match target regularity. A fractal dimension regularizer provides interpretable complexity control whose learned values recover the true fractal dimension of each target. These results establish regularity-matched basis design as a principled strategy for neural function approximation.

Figures & tables

Figure 2: FI-KAN: learning basis geometry across the smooth-to-rough spectrum. Top row: FI-KAN combines smooth basis functions (B-spline/hat-type), fractal interpolation with learnable contraction parameters d\mathbf{d} , and the KAN edge-function graph to produce regularity-matched edge geometry. Middle row: Smooth-to-fractal basis morphing as did_{i} increases from 00 to 0.90.9 . At di=0d_{i}=0 the basis is a smooth hat function ( dim⁡B=1\dim_{B}=1 ); as did_{i} increases, the basis acquires progressively finer self-affine structure with dim⁡B>1\dim_{B}>1 . Bottom row: The two FI-KAN variants. Pure FI-KAN (Barnsley framework) uses all-fractal bases, strongest on rough targets. Hybrid FI-KAN (Navascués framework) retains a spline backbone and adds a fractal correction, providing robustness across both smooth and rough regimes.
Figure 3: Edge function architecture for the three models. (a) KAN : base activation plus B-spline path. (b) Pure FI-KAN (Barnsley): replaces B-splines with fractal interpolation bases φm(x;d)\varphi_{m}(x;\mathbf{d}) . When d=0\mathbf{d}=\mathbf{0} , the FIF bases reduce to piecewise linear hat functions. (c) Hybrid FI-KAN (Navascués): retains the B-spline path and adds a fractal correction, implementing the α\alpha -fractal decomposition fbα=b+hf_{b}^{\alpha}=b+h . When d=0\mathbf{d}=\mathbf{0} and fractal weights vanish, Hybrid reduces to standard KAN.
Figure 4: Fractal basis function φ2(x;d)\varphi_{2}(x;\mathbf{d}) (the middle basis on a 5-interval grid) as the contraction parameters did_{i} vary uniformly from 0 to 0.9. At di=0d_{i}=0 , the basis is the standard piecewise linear hat function ( dim⁡B=1\dim_{\mathrm{B}}=1 ). As did_{i} increases, the basis acquires increasingly fine-scale fractal structure ( dim⁡B>1\dim_{\mathrm{B}}>1 ) while maintaining the Kronecker property φ2(xj)=δ2j\varphi_{2}(x_{j})=\delta_{2j} at all grid points (black dots). Recursion depth K=8K=8 .
Figure 5: Pure FI-KAN (Barnsley framework) architecture detail. (a) Network graph: all edges carry fractal interpolation function (FIF) bases φm(x;d)\varphi_{m}(x;\mathbf{d}) with learnable contraction parameters; nodes perform summation. (b) Edge activation: the input xx splits into a base SiLU path and a fractal path ∑wmfracφm(x;d)\sum w_{m}^{\mathrm{frac}}\varphi_{m}(x;\mathbf{d}) , where d∈(−1,1)N\mathbf{d}\in(-1,1)^{N} controls the fractal character of the basis. When d=0\mathbf{d}=\mathbf{0} , the FIF bases reduce to piecewise linear hat functions (order-1 KAN). (c) Fractal basis φ2(x;d)\varphi_{2}(x;\mathbf{d}) at four contraction values di∈{0,0.3,0.6,0.9}d_{i}\in\{0,0.3,0.6,0.9\} , computed via K=10K=10 truncated Read–Bajraktarević iterations. The Kronecker property φ2(xj)=δ2j\varphi_{2}(x_{j})=\delta_{2j} is preserved at all dd values (dots). (d) Box-counting dimension dim⁡B(d)\dim_{B}(\mathbf{d}) as a function of uniform did_{i} , with the transition surface ∑∣di∣=1\sum|d_{i}|=1 marked. (e) Fractal dimension regularizer R(d)=(dim⁡B(d)−1)2\mathcal{R}(\mathbf{d})=(\dim_{B}(\mathbf{d})-1)^{2} : a geometry-aware Occam’s razor that penalizes unnecessary fractal complexity.
Figure 6: Hybrid FI-KAN (Navascués framework) architecture detail. (a) Network graph: edges carry dual paths—solid blue for the B-spline path bb (classical approximant) and dashed coral for the fractal correction hh —implementing the α\alpha -fractal decomposition fbα=b+hf_{b}^{\alpha}=b+h . (b) Edge activation: the input splits into three paths (base SiLU, B-spline, FIF), combined into the α\alpha -fractal output fbα=b+hf_{b}^{\alpha}=b+h . When d=0\mathbf{d}=\mathbf{0} and wfrac=0\mathbf{w}^{\mathrm{frac}}=\mathbf{0} , the architecture reduces exactly to standard KAN. Initialization sets wfrac=0w^{\mathrm{frac}}=0 and draw∼N(0,0.01)d^{\mathrm{raw}}\sim\mathcal{N}(0,0.01) , so the network starts as a KAN and develops fractal structure only where the data demands it. (c) α\alpha -fractal decomposition showing the B-spline basis bb (blue), hybrid outputs b+hb+h at increasing did_{i} , and the isolated fractal correction hh (dashed). (d) Recovery property: ∥d∥→0\|\mathbf{d}\|\to 0 implies fbα→bf_{b}^{\alpha}\to b (KAN recovery). (e) Fractal energy ratio ρ=∥wfrac∥1/∥wspl∥1\rho=\|w^{\mathrm{frac}}\|_{1}/\|w^{\mathrm{spl}}\|_{1} across target Hölder exponents: fractal path is active ( ρ≫0\rho\gg 0 ) on rough targets and dormant ( ρ≈0\rho\approx 0 ) on smooth targets.
  Pure FI-KAN experimentHybrid FI-KAN experiment
Targetdim⁡B\dim_{\mathrm{B}}MLPKANPureMLPKANHybrid
polynomial1.001.42e-22.97e-33.20e-11.44e-22.97e-31.3e-5
exp_sin1.002.18e-13.32e-44.41e-11.68e-13.32e-47.0e-6
chirp1.004.51e-12.95e-13.34e-14.51e-12.95e-11.57e-1
weierstrass std{}_{\text{std}}1.642.20e-11.23e-11.01e-12.20e-11.23e-14.38e-2
weierstrass rough{}_{\text{rough}}1.684.48e-11.98e-12.15e-14.48e-11.98e-19.31e-2
Table 1: 1D regression benchmark (test MSE, mean ±\pm std over 5 seeds, grid size G=8G=8 ). Pure FI-KAN parameters: MLP=487, KAN=416, FI-KAN=488. Hybrid FI-KAN parameters: MLP=841, KAN=416, FI-KAN=840. Best result per target shown in bold .
Figure 7: Hybrid FI-KAN vs. KAN on four representative targets spanning the regularity spectrum. Top row: target function (gray), KAN fit (blue dashed), and Hybrid FI-KAN fit (red) with final test MSE. Bottom row: training curves (test MSE vs. epoch, log scale) with improvement ratios. On smooth targets (polynomial, exp_sin), Hybrid FI-KAN converges to orders-of-magnitude lower MSE. On fractal targets (Weierstrass, sawtooth), the fractal correction path captures multi-scale structure that B-splines miss entirely.
α\alphaMLPKANHybrid FI-KANImprovement over KAN
0.27.42e-13.99e-13.02e-11.3×1.3\times
0.44.59e-11.69e-16.77e-22.5×2.5\times
0.62.94e-17.36e-21.20e-26.1×6.1\times
0.81.94e-13.14e-22.34e-313.4×13.4\times
1.01.34e-11.32e-26.30e-421.0×21.0\times
1.56.99e-21.58e-34.8e-532.9×32.9\times
Table 2: Hölder regularity sweep (test MSE, Hybrid FI-KAN experiment). Hybrid FI-KAN wins at every regularity level.
Figure 8: Hölder regularity sweep: test MSE as a function of Hölder exponent α\alpha for fα(x)=∣x∣αf_{\alpha}(x)=|x|^{\alpha} . Hybrid FI-KAN achieves the lowest MSE at every regularity level, with improvement factors over KAN increasing from 1.3×1.3\times ( α=0.2\alpha=0.2 ) to 33×33\times ( α=1.5\alpha=1.5 ). The three curves are clearly separated across the entire spectrum, with the C1C^{1} boundary ( α=1\alpha=1 ) marked. Shaded regions show ±1\pm 1 std over 5 seeds.
TargetKAN (864p)Hybrid FI-KAN (840p)RatioWinner
polynomial1.58e-41.2e-513.2×13.2\timesFI-KAN
exp_sin4.0e-66.0e-60.7×0.7\timesKAN
weierstrass std{}_{\text{std}}4.58e-24.24e-21.1×1.1\timesFI-KAN
sawtooth4.46e-31.81e-32.5×2.5\timesFI-KAN
Table 3: Parameter-matched comparison (Hybrid FI-KAN: 840 params vs. KAN: 864 params). FI-KAN wins 3 of 4 targets; the one KAN win (exp_sin) is nearly tied.
 Test MSE at grid size GGScaling exponent γ\gamma
Target358121620MLPKANFI-KAN
Hybrid FI-KAN, smooth target: exp_sin
MLP2.29e-12.02e-11.68e-11.08e-16.82e-24.16e-20.583.210.18
KAN5.91e-23.10e-33.32e-49.1e-53.9e-53.3e-5
FI-KAN9.0e-51.5e-56.0e-66.0e-61.2e-41.5e-5
Hybrid FI-KAN, fractal target: sawtooth ( dim⁡B=1.5\dim_{\mathrm{B}}=1.5 )
Table 4: Scaling laws: MSE vs. parameter count at varying grid sizes (Hybrid FI-KAN). Exponent γ\gamma fitted via log-log regression.
Figure 9: Scaling laws (Hybrid FI-KAN): test RMSE vs. parameter count across three targets spanning the regularity spectrum. Left (exp_sin, smooth): FI-KAN achieves low error from the smallest grid size, saturating early while KAN continues improving with more parameters. Center (Weierstrass, rough): FI-KAN achieves lower RMSE than KAN at all model sizes, with both showing positive scaling. Right (sawtooth, fractal): MLP completely stagnates. FI-KAN maintains a consistent advantage over KAN at every model size. Scaling exponents fitted over the full grid range G∈{3,5,8,12,16,20}G\in\{3,5,8,12,16,20\} are reported in Table 4 . Shaded regions show ±1\pm 1 std over 5 seeds.
 Weierstrass ( dim⁡B≈1.64\dim_{\mathrm{B}}\approx 1.64 )Sawtooth ( dim⁡B=1.5\dim_{\mathrm{B}}=1.5 )
SNR (dB)KANHybrid FI-KANKANHybrid FI-KAN
100.01.567e-14.52e-21.139e-21.87e-3
40.01.568e-14.62e-21.140e-21.85e-3
30.01.569e-14.62e-21.141e-21.84e-3
20.01.573e-14.74e-21.144e-21.92e-3
10.01.579e-15.17e-21.155e-22.13e-3
Table 5: Noise robustness (test MSE on clean data after training with noisy targets). Both FI-KAN variants outperform KAN across all noise levels on fractal targets.
Figure 10: Noise robustness on fractal targets (Hybrid FI-KAN). Models trained with noisy data, evaluated on clean test data. Left (Weierstrass): FI-KAN maintains a ∼ ⁣3.5×\sim\!3.5\times advantage across all SNR levels, with the gap narrowing gracefully under heavy noise. Right (sawtooth): up to 6.1×6.1\times advantage at clean SNR, degrading gracefully to 4.7×4.7\times at extreme noise ( 55 dB). Shaded regions show ±1\pm 1 std over 5 seeds.
ModelFinal MSEStdRelative to KAN
MLP1.2940.0081.04×1.04\times worse
KAN1.2480.022(baseline)
Pure FI-KAN1.1580.0511.1×1.1\times better
Hybrid FI-KAN1.3760.0351.1×1.1\times worse
Table 6: Continual learning: final MSE after sequential training on multiple tasks.
Figure 11: Continual learning across 5 sequential tasks (Hybrid FI-KAN). Each column shows one training phase; each row shows a different architecture’s output. Top (Data): the current task’s target function (black) with all tasks shown in gray. MLP: catastrophic forgetting collapses the output to a near-constant function. KAN: retains some structure but progressively distorts earlier tasks. FI-KAN: retains peaked structure across phases, with the fractal and spline paths providing implicit modularity. Quantitative MSE for the task sequence including the corrected Takagi–Landsberg target is reported in Table 6 .
 Polynomial ( dim⁡B=1\dim_{\mathrm{B}}=1 )Weierstrass ( dim⁡B≈1.64\dim_{\mathrm{B}}\approx 1.64 )
λfrac\lambda_{\mathrm{frac}}Test MSELearned dim⁡B\dim_{\mathrm{B}}Test MSELearned dim⁡B\dim_{\mathrm{B}}
03.74e-11.3029.68e-21.150
10−410^{-4}3.69e-11.2999.63e-21.146
10−310^{-3}3.41e-11.2949.15e-21.137
10−210^{-2}1.23e-11.2317.28e-21.053
10−110^{-1}9.27e-31.0016.09e-21.008
Table 7: Regularization sweep (Pure FI-KAN). The regularizer drives the learned fractal dimension toward 1.0 and significantly improves performance on the smooth target.
Figure 12: Regularization sweep (Hybrid FI-KAN) on polynomial (left) and Weierstrass (right) targets. Left (polynomial): test MSE is nearly flat across λfrac\lambda_{\mathrm{frac}} , confirming that the spline path already handles smooth targets effectively; the fractal regularizer has little to correct. Right (Weierstrass): stronger regularization ( λfrac≥10−2\lambda_{\mathrm{frac}}\geq 10^{-2} ) reduces MSE by ∼ ⁣20%\sim\!20\% , suppressing unnecessary fractal complexity while preserving beneficial multi-scale structure. Shaded regions show ±1\pm 1 std. The Hybrid variant is robust to regularization weight choice, in contrast to the Pure variant’s sensitivity ( Table 7 ).
Depth KKTest MSEStdWall time (s)
13.39e-28.35e-45.8
22.72e-28.33e-47.1
43.03e-27.95e-49.6
64.11e-21.45e-312.0
85.43e-23.63e-314.4
106.26e-23.87e-316.9
Table 8: Fractal depth analysis (Hybrid FI-KAN, Weierstrass target). K=2K=2 is optimal; deeper recursion degrades performance and increases training time.
Figure 13: Impact of fractal recursion depth KK on Hybrid FI-KAN performance (Weierstrass target). Red (left axis): test MSE (log scale) with ±1\pm 1 std error bars. Blue dashed (right axis): average training time. K=2K=2 achieves optimal MSE at minimal computational cost. Deeper recursion ( K>4K>4 ) monotonically degrades performance while linearly increasing training time, reflecting the optimization difficulty of propagating gradients through multiple sequential multiplicative steps.
TargetMLPKANPure FI-KANHybrid FI-KAN
Ackley 2D4.27e-12.56e-26.03e-21.75e-3
Weierstrass 2D4.32e-11.87e-11.25e-18.95e-2
Table 9: 2D function regression (test MSE, mean ±\pm std over 5 seeds).
Figure 14: 2D function regression (Hybrid FI-KAN). Top row (Ackley): KAN smooths out the multi-modal structure near the origin (MSE 0.0204), while FI-KAN recovers the peaked landscape (MSE 0.0016), a 12.8×12.8\times improvement. Bottom row (2D Weierstrass): KAN captures only coarse structure (MSE 0.1577), while FI-KAN resolves finer oscillatory detail (MSE 0.0826), a 1.9×1.9\times improvement. Left column: target surface. Center: KAN prediction. Right: Hybrid FI-KAN prediction.
ProblemRoughnessMLPKANHybrid FI-KANImprov.Source
Rough-coefficient diffusion −ddx(a(x)dudx)=1-\frac{d}{dx}(a(x)\frac{du}{dx})=1 , a(x)=e0.5BH(x)a(x)=e^{0.5B_{H}(x)}
Hc=0.1H_{c}=0.1rough coeff2.20e-31.26e-31.74e-573×73\timesFEM
Hc=0.3H_{c}=0.3rough coeff3.05e-31.19e-31.50e-579×79\timesFEM
Hc=0.5H_{c}=0.5rough coeff2.11e-31.15e-31.50e-577×77\timesFEM
Hc=0.7H_{c}=0.7smooth coeff8.39e-41.11e-31.71e-565×65\timesFEM
L-shaped domain −Δu=1-\Delta u=1 , corner singularity u∼r2/3sin⁡(2θ/3)u\sim r^{2/3}\sin(2\theta/3)
Table 10: Non-smooth PDE solutions (test MSE, mean ±\pm std over 5 seeds). Reference solutions computed via scikit-fem and exact spectral methods. Hybrid FI-KAN wins all 12 experiments. The rough-coefficient diffusion results ( 6565 – 79×79\times ) are the strongest in the paper.
Figure 15: L-shaped domain benchmark ( −Δu=1-\Delta u=1 , Hölder 2/32/3 corner singularity). Left: FEM reference solution computed via scikit-fem (P1 elements, 6144 elements); the singularity at the re-entrant corner is visible as the peaked region near the origin. Center: Test MSE comparison (log scale, ±1\pm 1 std over 5 seeds). Hybrid FI-KAN achieves 3.5×3.5\times lower MSE than KAN. Right: Training curves showing Hybrid FI-KAN converging to ∼ ⁣10−3\sim\!10^{-3} MSE while KAN plateaus near 3×10−33\times 10^{-3} .
TargetTrue α\alphaPure dim⁡B\dim_{\mathrm{B}}Hybrid dim⁡B\dim_{\mathrm{B}}
smooth ( α=2.0\alpha=2.0 )2.01.128±0.0121.128\pm 0.0121.000±0.0001.000\pm 0.000
C1C^{1} ( α=1.0\alpha=1.0 )1.01.157±0.0121.157\pm 0.0121.034±0.0071.034\pm 0.007
Hölder 0.60.61.194±0.0141.194\pm 0.0141.119±0.0071.119\pm 0.007
Hölder 0.30.31.190±0.0261.190\pm 0.0261.152±0.0181.152\pm 0.018
Weierstrass0.641.144±0.0081.144\pm 0.0081.149±0.0041.149\pm 0.004
Table 11: Learned fractal dimension (mean ±\pm std over 5 seeds) for targets of varying regularity. The Hybrid variant produces sharper estimates, with smooth targets yielding dim⁡B≈1\dim_{\mathrm{B}}\approx 1 and rough targets yielding dim⁡B>1.1\dim_{\mathrm{B}}>1.1 .
Figure 16: Learned fractal dimension vs. true Hölder exponent (Hybrid FI-KAN). The learned dim⁡B\dim_{\mathrm{B}} correlates monotonically with target regularity: smooth targets ( α=2.0\alpha=2.0 ) yield dim⁡B=1.0\dim_{\mathrm{B}}=1.0 exactly (fractal path inactive), while rough targets ( α=0.3\alpha=0.3 , Weierstrass) yield dim⁡B>1.1\dim_{\mathrm{B}}>1.1 (fractal path active). The dotted line at dim⁡B=1\dim_{\mathrm{B}}=1 marks the smooth baseline. This monotonic correlation makes the learned fractal dimension a useful, interpretable proxy for target regularity.
Figure 17: Fractal parameter evolution during training (Pure FI-KAN). Top row: individual did_{i} trajectories for layer 0 across three targets. Bottom row: mean dim⁡B\dim_{\mathrm{B}} over training. Left (polynomial): the contraction parameters grow steadily, pushing dim⁡B\dim_{\mathrm{B}} above 1.2; this is the smooth-target overfitting pathology predicted by Corollary 4.8 . Center (Weierstrass): dim⁡B\dim_{\mathrm{B}} stays near 1.0 throughout, indicating the regularizer successfully constrains fractal complexity. Right (sawtooth): dim⁡B\dim_{\mathrm{B}} spikes to ∼ ⁣1.05\sim\!1.05 in early training (exploration phase) then settles back, suggesting an explore-then-consolidate dynamic where the network first probes fractal structure and then retains only what reduces the loss.
Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

 Mean MSEParams
TargetMLPKANHybridMLPKANHybrid
J0J_{0} Bessel8.62e-21.54e-42.18e-4424208424
exsin⁡(y)+y2e^{x}\sin(y)+y^{2}7.79e-14.32e-26.16e-2609312608
x⋅yx\cdot y4.6e-58.3e-58.2e-5609312608
10D sum4.57e-31.21e-24.03e-410815721080
4D composite2.501.384.27639338652
Appendix
Table 12: KAN paper toy functions [ 27 ] . These are smooth, low-dimensional targets where B-spline KAN is near-optimal. Hybrid FI-KAN wins 1 of 5; Pure FI-KAN wins 0 of 5. This is consistent with the theoretical analysis: fractal bases provide no advantage on smooth analytic targets.
TargetMLPKANPure FI-KAN
J0J_{0} Bessel8.62e-21.54e-47.30e-3
exsin⁡(y)+y2e^{x}\sin(y)+y^{2}7.93e-14.32e-21.98
x⋅yx\cdot y6.1e-58.3e-58.22e-4
10D sum8.17e-31.21e-23.05e-1
4D composite2.751.389.59
Appendix
Table 13: KAN paper toy functions (Pure FI-KAN).
Figure 18: KAN paper toy functions (Hybrid FI-KAN). Bar chart comparison of test MSE across five canonical smooth targets. KAN wins on 3 of 5 targets where B-splines are near-optimal. The one Hybrid FI-KAN win (10D sum, ∼ ⁣30×\sim\!30\times improvement) suggests that the fractal correction provides additional representational diversity that benefits higher-dimensional targets even when the underlying function is smooth.
EquationMLPKANHybrid FI-KANBest
I.6.2 Gaussian6.46e-11.32e-29.77e-3FI-KAN
I.12.11 Lorentz9.40e-42.15e-24.53e-2MLP
I.16.6 Relativity8.27e-27.35e-31.97e-3FI-KAN
I.29.16 Distance2.00e-22.15e-32.92e-3KAN
I.30.3 Diffraction2.10e-11.07e-15.45e-2FI-KAN
I.50.26 Cosine7.33e-21.12e-21.33e-2KAN
Appendix
Table 14: Feynman physics equations (Hybrid FI-KAN). FI-KAN wins 3 of 7 on these smooth physics targets.
Figure 19: Feynman physics equations (Hybrid FI-KAN). Bar chart comparison across seven analytic physics targets. KAN wins 4 of 7, consistent with the expectation that B-splines are well-adapted to analytic targets. The FI-KAN wins (Gaussian, relativity, diffraction) occur on functions with sharp transitions or oscillatory structure that benefit from multi-scale representation.
Figure 20: Scaling laws (Pure FI-KAN): test RMSE vs. parameter count. Left (exp_sin): Pure FI-KAN exhibits a negative scaling exponent: test error increases with model size. This is the pathological behavior predicted by Corollary 4.8 : additional grid points introduce more oscillatory fractal basis functions, making the cancellation required for smooth approximation increasingly fragile. Center (Weierstrass): mild negative scaling. Right (sawtooth): negative scaling at large grid sizes, extending the smooth-approximation obstruction to fractal targets when too many uncontrolled fractal bases are introduced without the spline backbone. At small-to-moderate grid sizes, positive scaling is observed before the pathology sets in.
Figure 21: Pure FI-KAN vs. KAN on four representative targets. Top row: function fits. Bottom row: training curves with improvement ratios. The pattern is complementary to Hybrid FI-KAN ( Fig. 7 ): Pure FI-KAN loses badly on smooth targets (ratio 0.0x on polynomial, exp_sin) but wins on the sawtooth (ratio 2.3x). This asymmetry is the strongest controlled evidence for the regularity-matching hypothesis.
Figure 22: Noise robustness (Pure FI-KAN). The Pure variant also outperforms KAN on fractal targets under noise, though with smaller margins than the Hybrid. Left (Weierstrass): ∼ ⁣1.5×\sim\!1.5\times advantage. Right (sawtooth): up to 2.2×2.2\times advantage at clean SNR.
Figure 23: Learned fractal dimension vs. true Hölder exponent (Pure FI-KAN). The monotonic correlation is present but weaker than the Hybrid variant ( Fig. 16 ): smooth targets receive dim⁡B≈1.13\dim_{\mathrm{B}}\approx 1.13 rather than 1.0, because the Pure variant is forced to use some fractal structure even when inappropriate. This systematic overestimation further confirms that fractal-only bases are suboptimal for smooth targets.
Figure 24: 2D function regression (Pure FI-KAN). Pure FI-KAN loses on Ackley (smooth outside origin) but captures more fine-scale structure on the 2D Weierstrass target than KAN (MSE 0.1251 vs. 0.1871), consistent with the regularity-matching thesis.

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