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ER-KANs: Efficient and Robust Kolmogorov-Arnold Networks for Data-Scarce Scientific Machine Learning
Harshil Lodhiya
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
The paper introduces ER-KAN, a variant of Kolmogorov-Arnold Networks (KANs) designed for robustness in noisy, data-scarce scientific machine learning settings. ER-KAN utilizes shared Gaussian Radial Basis Function (RBF) bases, curriculum noise injection, and entropy-weighted adaptive regularization. Experiments demonstrate that while ER-KAN is less accurate than ChebyKAN on clean data, it exhibits significantly lower noise degradation (1.4x vs 10.6x for ChebyKAN at sigma=0.1) and outperforms MLPs in specific physics-informed neural network (PINN) tasks like the damped harmonic oscillator.
Entities (10)
Relation Signals (7)
ER-KAN → appliedto → Damped Harmonic Oscillator
confidence 95% · evaluate on a damped harmonic oscillator physics-informed neural network
ER-KAN → appliedto → Burgers' Equation
confidence 95% · evaluate on a Burgers’ equation PINN where all models fail to converge
ER-KAN → haslowernoisedegradationthan → ChebyKAN
confidence 95% · ER-KAN degrades 1.4x... versus 10.6x for ChebyKAN
ER-KAN → usesbasisfunction → Gaussian RBF
confidence 95% · ER-KAN combines... shared Gaussian RBF bases across all edges in a layer
ChebyKAN → usesbasisfunction → Chebyshev Polynomials
confidence 95% · ChebyKAN computes degree-d Chebyshev polynomials
ChebyKAN → hashighernoisedegradationthan → Vanilla KAN
confidence 90% · ChebyKAN’s test MSE... increases by a factor of 10.6x... versus 7.9x for vanilla KAN
ER-KAN → outperforms → MLP
confidence 90% · ER-KAN achieves 4.2x lower solution MSE than MLP [on damped harmonic oscillator]
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Abstract
Abstract:The efficient-KAN literature---covering Chebyshev, wavelet, and radial-basis-function variants of the original Kolmogorov-Arnold Network---has been benchmarked almost entirely on clean data. We show that this choice conceals a large capability difference between architectures: ChebyKAN's test MSE (evaluated against clean ground truth) increases by a factor of 10.6x when training data is corrupted with sigma=0.1 noise, versus 7.9x for vanilla KAN, 1.7x for a standard MLP, and just 1.4x for our proposed ER-KAN. ER-KAN combines three design choices targeting the noisy, data-scarce setting: shared Gaussian RBF bases across all edges in a layer (providing locality and efficient parameterisation), curriculum noise injection during training (explicitly teaching noise robustness), and entropy-weighted adaptive regularisation (preventing overfitting at small N). The result is a 595-parameter network that matches MLP accuracy at moderate noise while degrading far more gracefully as noise grows. We evaluate on eight analytic functions (N in {50, 200, 500}, sigma in {0, 0.03, 0.1}), on a damped harmonic oscillator physics-informed neural network where ER-KAN achieves 4.2x lower solution MSE than MLP, and on a Burgers' equation PINN where all models fail to converge---a genuine limitation we report rather than suppress. We introduce the noise degradation ratio as a simple complementary metric and recommend it become a standard reporting requirement for efficient-KAN papers.
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- Source: https://arxiv.org/abs/2608.14773v1
- Canonical: https://arxiv.org/abs/2608.14773v1
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ER-KANs: Efficient and Robust Kolmogorov–Arnold Networks for Data-Scarce Scientific Machine Learning Harshil Lodhiya hlodhiya@slicedhealth.com Sliced Health Abstract The efficient-KAN literature—covering Chebyshev, wavelet, and radial-basis-function variants of the original Kolmogorov–Arnold Network—has been benchmarked almost entirely on clean data. We show that this choice conceals a large capability difference between architectures: ChebyKAN’s test MSE (evaluated against clean ground truth) increases by a factor of 10.6× when training data is corrupted withσ= 0.1 noise, versus 7.9×for vanilla KAN, 1.7×for a standard MLP, and just 1.4× for our proposed ER-KAN. ER-KAN combines three design choices targeting the noisy, data-scarce setting: shared Gaussian RBF bases across all edges in a layer (providing locality and efficient parameterisation), curriculum noise injection during training (explicitly teaching noise robustness), and entropy- weighted adaptive regularisation (preventing overfitting at smallN). The result is a 595-parameter network that matches MLP accuracy at moderate noise while degrading far more gracefully as noise grows. We evaluate on eight analytic functions (N∈50,200,500,σ∈0,0.03,0.1), on a damped harmonic oscillator physics-informed neural network where ER-KAN achieves 4.2×lower solution MSE than MLP, and on a Burgers’ equation PINN where all models fail to converge—a genuine limitation we report rather than suppress. We introduce the noise degradation ratio as a simple complementary metric and recommend it become a standard reporting requirement for efficient-KAN papers. 1 Introduction Kolmogorov–Arnold Networks [Liu et al., 2024] arrived with an appealing pitch: replace fixed, node- based activations with learnable edge functions, gaining interpretability without losing expressive power. The original B-spline implementation validated this on small benchmarks, but a forward pass through a full B-spline layer is slow—each edge requires evaluating a spline over a grid at every call. Within months, multiple groups proposed faster basis functions: Chebyshev polynomials [SynodicMonth, 2024], wavelets [Bozorgasl and Chen, 2024], reflective linear functions [Noutsos and Roumeliotis, 2024], and Gaussian RBFs [Li, 2024]. All of them benchmarked on clean data. Scientific machine learning rarely has clean data. Sensor noise, simulation discretisation error, and experimental uncertainty routinely corrupt training targets by 5–20%. And data is often scarce: building a surrogate model from a handful of expensive simulation runs is standard practice in computational biology, materials science, and geophysics [Willard et al., 2022]. In this setting, the choice of basis function turns out to matter—a lot. Our central finding is summarised in Figure 2 and Table 2: Chebyshev polynomials, despite being the best approximators on clean data (geomean RMSE 0.025 vs 0.030–0.132 for all other models atN=50), amplify the effect ofσ=0.1 noise by a factor of 10.6×. Vanilla B-spline KAN 1 arXiv:2608.14773v1 [cs.LG] 14 Aug 2026 degrades 7.9×. ER-KAN’s Gaussian RBF basis degrades only 1.4×—close to a parameter-matched MLP’s 1.7× and by far the best among KAN-family models. ER-KAN is not the uniformly best model in our experiments. On clean data, Chebyshev and vanilla KAN substantially outperform it; even atσ=0.1, ChebyKAN still achieves lower absolute RMSE (0.083 vs 0.158 for ER-KAN) because it started from such a low clean baseline. The case for ER-KAN is therefore not “use this everywhere” but rather: in any setting where noise level is uncertain, deployment will see higher noise than training, or physics-informed learning removes labeled data entirely—ER-KAN is the safer architectural choice. Contributions. •We introduce ER-KAN, a lightweight KAN variant (595 params) combining shared Gaussian RBF bases, curriculum noise injection, and adaptive regularisation. •We run the first systematic noisy-data comparison of four KAN-family architectures plus MLP: 8 functions × 3 sample sizes × 3 noise levels × 5 seeds = 480 runs per model. •We introduce the noise degradation ratioD R as a complementary evaluation metric and show it reveals a 7-fold capability difference invisible to clean-data RMSE. •We report two PINN experiments with opposite outcomes—ER-KAN wins on a smooth oscillator, all models fail on a shock-dominated Burgers’ equation—and explain why. 2 Background and Related Work 2.1 Kolmogorov–Arnold Networks Liu et al. [2024] introduced KANs by placing learnable univariate functions on the edges of a network, following the Kolmogorov–Arnold representation theorem [Kolmogorov, 1957, Sprecher, 1965]. Each edge function φ ij is parameterised as a linear combination of B-spline basis functions: φ ij (x) = w b b(x) + X k c k B k (x),(1) whereb(x) is a residual SiLU activation,B k are cubic B-spline basis functions on a fixed grid, and c k ,w b are trainable. The layer outputh (l+1) j = P i φ ij (h (l) i ) replaces the standard dot-product-plus- activation of an MLP. 2.2 Efficient KAN Variants The B-spline evaluation in Eq.(1)is expensive because it requires looking up grid values for every sample, every edge, every layer. Efficient variants replace this with analytically cheaper bases: FastKAN [Li, 2024] uses fixed Gaussian RBF centres—the closest prior to ER-KAN. The key difference is that FastKAN assigns per-edge centre sets while ER-KAN shares one set of centres across all edges in a layer, halving the parameter count and (as the ablation shows, Table 7) substantially reducing noise sensitivity. ChebyKAN [SynodicMonth, 2024] computes degree-dChebyshev polynomials via the three- term recurrenceT n = 2xT n−1 −T n−2 , requiring onlyO(d) multiplications per element. It is fast and accurate on clean data, but we show its derivative bound|T ′ d (x)|≤ d 2 leads to input-amplification of perturbations proportional to d 2 . WaveKAN [Bozorgasl and Chen, 2024] and FasterKAN [Noutsos and Roumeliotis, 2024] use wavelet and reflective linear bases respectively; we do not include them in our primary comparison but the degradation ratio framework applies directly. 2 2.3 Physics-Informed Neural Networks PINNs [Raissi et al., 2019] encode a governing PDE as a residual loss, enabling training without labeled solution data. Wang et al. [2022] show that spectral bias—MLPs preferentially fitting low-frequency components—is a key failure mode. KAN-based PINNs have been explored by Shukla et al. [2024], with results that are problem-dependent; our experiments confirm this. 2.4 Noise Robustness in Function Approximation The sensitivity of polynomial interpolation to input perturbations is well-understood theoretically (Runge phenomenon, Chebyshev stability analysis). In the neural network literature, noise robustness is typically studied through data augmentation [Simard et al., 1998] or input dropout [Srivastava et al., 2014]. To our knowledge, no prior work has compared KAN basis functions specifically through the lens of noise degradation. 3 ER-KAN Architecture 3.1 Motivation: Why RBFs Degrade Less Under Noise For a Chebyshev basis of degreed, the classical bound|T ′ d (x)| ≤ d 2 means that a perturbation εin input space can produce an output change as large asw d d 2 εfor a single term of weightw d . Ford=8 (our ChebyKAN baseline), high-degree terms have derivatives up to 64 times the input perturbation. For a Gaussian RBF basis functionφ g (x) =exp(−(x− c g ) 2 / (2σ 2 g )), the derivative is|φ ′ g (x)|= |x− c g |/σ 2 g · φ g (x)≤1/(σ g √ e). For our defaultσ g = 0.1, this is bounded by approximately 3.7—an order of magnitude smaller than the degree-8 Chebyshev bound. This analytic argument predicts lower noise sensitivity for RBF bases, and the empirical degradation ratios (Table 2) confirm it. 3.2 Shared Gaussian RBF Basis For a layer withn in inputs andn out outputs, we placeGGaussian RBF centresc g G g=1 uniformly in [−1,1]. These centres are shared across alln in × n out edges; each edge has onlyGscalar weights. The j-th unit output is: h j = n in X i=1 G X g=1 w ijg φ g (x i ) + b j , φ g (x) = exp − (x− c g ) 2 2σ 2 g ! ,(2) whereσ g are trainable widths (initialised to 1/G). Ablation shows this sharing is the single most important component of ER-KAN: without it, the geomean MSE ratio degrades to 1.54×the full model (Table 7). 3.3 Curriculum Noise Injection We perturb training inputs at epoch e with: ̃ x = x + ε, ε∼N(0, σ 2 e I), σ e = σ base 1− e E 2 ,(3) whereσ base matches the expected noise level andEis total epochs. The quadratic decay provides a smooth transition from aggressive augmentation (large-scale structure first) to clean training (fine-tuning). Removing this component increases geomean MSE by 7% (Table 7). 3 3.4 Adaptive Regularisation We penalise edge weights with an entropy-weighted ℓ 1 term: R(W) = λ X i,j H ij ·∥w ij· ∥ 1 ,(4) whereH ij is the activation entropy of edge (i,j) over the current batch, encouraging sparse activations on small datasets. Interestingly, the ablation shows this component has no measurable effect in our setting (1.00×ratio); we retain it as a regularisation safeguard but do not claim it as a contributing factor. 3.5 Architecture and Parameter Counts Table 1 compares parameter counts and inference latency. ER-KAN uses 2 hidden layers of width 32 withG=8 RBF centres, yielding 595 parameters. The 1D and 2D input variants differ only in the first-layer parameter count. Table 1: Parameter counts and per-epoch training time on CPU (500 observations, 5 seeds). Inference latency in microseconds per sample. ModelParams (1D) Params (2D) Training (s) Inf. (μs/samp.) MLP4,3534,417 0.144± 0.111.07± 0.43 ER-KAN595852 0.296± 0.021.76± 0.12 Vanilla KAN801– 0.799± 0.038.02± 0.37 ER-KAN trains 2.7×faster than vanilla KAN per epoch and has 8×lower inference latency. It is slower than MLP (2×per epoch), which is expected given the Gaussian evaluation overhead; the inference gap (1.76 vs 1.07 μs) is negligible in practice. 3.6 Training Protocol We train with Adam [Kingma and Ba, 2015], cosine learning rate from 10 −3 to 10 −5 , batch size 64, early stopping (patience 500 epochs) on a 20% held-out validation split. For PINNs, no labeled data split is used; instead we train for a fixed number of epochs with physics, initial condition, and boundary condition losses. Full pseudocode is in Algorithm 1. Algorithm 1 ER-KAN training Require: Data (x n ,y n ), σ base , epochs E 1: for e = 1 to E do 2: σ e ← σ base (1− e/E) 2 3: for each batch B do 4:Perturb: ̃ x = x +N(0,σ 2 e ) 5:Loss: L = MSE(f θ ( ̃ x),y) +R(θ) 6:Adam step on θ 7: end for 8: Validate; update best checkpoint 9: end for 4 4 Experimental Setup 4.1 Analytic Function Suite We test on eight functions:sin(πx), the Runge function 1/(1 + 25x 2 ),|x|,xe −3x (Gauss-cosine enve- lope), a step function,xe −3x sin(3x), and two 2D functionsx 2 1 +x 2 2 (quadratic) andsin(πx 1 )cos(πx 2 ). These span oscillatory, algebraic, smooth-exponential, and discontinuous-like behaviours. Training inputs are drawn uniformly from [−1,1] d ; test inputs are a fixed 2,000-point grid. Test labels are always clean (no noise); only training labels are corrupted. We useN ∈50,200,500, σ ∈0,0.03,0.1, and 5 seeds, giving 8×3×3×5 = 360 runs per model (1,440 total across four models). 4.2 Noise Degradation Ratio For model m, function f, and sample count N, we define: D R (m,f,N,σ) = MSE(m,f,N,σ) MSE(m,f,N, 0) ,(5) the multiplicative increase in clean-test MSE when training onσ-noisy data relative to training on clean data. We report the geometric mean ofD R across the eight functions at fixedNandσ. A model withD R ≈1 is insensitive to training noise;D R ≫1 means the model is not just slower to converge but qualitatively impacted by noise in a way that compounds across the function suite. 4.3 PINN Experiments Damped harmonic oscillator. We solve: ̈x + 2ζω ̇x + ω 2 x = 0, x(0) = 1, ̇x(0) = 0, ζ = 0.15, ω = 2.0,(6) overt∈[0,10] using 200 collocation points, with loss weights 1:10:5 for physics, initial position, and initial velocity respectively. We train for 5,000 epochs. Burgers’ equation. We solve: u t + u x = ν u x , u(x, 0) =− sin(πx), u(±1,t) = 0, ν = 0.01 π ,(7) over [−1,1]×[0,1] with 2,500 interior collocation points and loss weights 1:20:20 for physics, IC, and BC. We train for 20,000 epochs. Reference solutions use SciPy RK45 on a 512×201 grid (rtol=10 −9 , atol=10 −11 ). 4.4 Baselines All experiments compare four models: ER-KAN (595 params, as described), ChebyKAN (320 params, degree 8, zero-initialised residual connection), vanilla KAN (801 params, cubic B-spline G= 5), and MLP (4,353 params for 1D, 8,577 for PINN, 2 hidden layers×64 units, Tanh activation). We also include the officialefficient-kanlibrary [Blealtan, 2024] as an external baseline in Section 5.5. All models use the same outer training loop; only ER-KAN uses curriculum noise augmentation. 5 MLP Vanilla KAN ER-KAN 0 1000 2000 3000 4000 Parameters MLP Vanilla KAN ER-KAN 3 × 10 −4 4 × 10 −4 6 × 10 −4 Clean MSE (mean) MLP Vanilla KAN ER-KAN 0 2 4 6 8 10 Training time (s) Single-oscillator benchmark: CPU vs MPS (5 seeds, 500 observations) MLPVanilla KANER-KANCPUMPS Figure 1: Initial benchmark (single-oscillator surrogate, 500 observations, clean data, 5 seeds). Vanilla KAN achieves the lowest MSE; ER-KAN and MLP are comparable. ER-KAN trains as fast as MLP and roughly 6×faster than vanilla KAN. Note: ChebyKAN was evaluated separately on the analytic function suite (Section 5.1) and does not appear here. 5 Results 5.1 Clean-Data Performance Figure 1 shows the initial three-model benchmark on a single-oscillator surrogate: vanilla KAN leads on MSE, ER-KAN and MLP are comparable, and ER-KAN trains roughly 6×faster than vanilla KAN. The analytic function suite brings in ChebyKAN and paints a fuller picture. AtN=50,σ=0, ChebyKAN achieves geomean RMSE 0.025 and vanilla KAN 0.030, while MLP reaches 0.119 and ER-KAN 0.132 (Table 3). This advantage holds across all sample sizes. We state this plainly because the contribution of this paper is not that ER-KAN is a better approximator on clean data. It is not. The contribution is that clean-data rankings are a misleading guide to behaviour under noise. 5.2 Noise Degradation Ratio Table 2 shows the geometric mean noise degradation ratio D R at N =50, σ=0.1. Table 2: Noise degradation ratio (geomean across 8 functions,N= 50): the factor by which clean-test MSE increases when training data hasσ=0.1 noise vs. clean training data. Lower is more noise-robust. ModelD R at σ=0.1 D R at σ=0.03 ER-KAN1.421.04 MLP1.661.05 Vanilla KAN7.882.48 ChebyKAN10.612.64 6 ER-KANMLP Vanilla KANChebyKAN 0.0 0.5 1.0 1.5 2.0 2.5 3.0 R (lower = more robust) 1.04× 1.05× 2.48× 2.64× σ = 0.03 No degradation (ratio = 1×) ER-KANMLP Vanilla KANChebyKAN 0 2 4 6 8 10 12 R (lower = more robust) 1.42× 1.66× 7.88× 10.61× σ = 0.1 No degradation (ratio = 1×) Noise degradation ratio R = MSE(σ)/MSE(0) — N=50, geomean across 8 analytic functions Figure 2: Noise degradation ratioD R for all four models atσ= 0.03 (left) andσ= 0.1 (right), N= 50, geomean across 8 analytic functions. ER-KAN and MLP remain close to 1.0 (robust); ChebyKAN degrades 10.6×and vanilla KAN 7.9×atσ= 0.1. The dashed line marks the “no degradation” baseline. The differences are large and systematic. Even at the moderateσ= 0.03 level, ChebyKAN degrades 2.6×—more than 50×worse than ER-KAN’s 1.04×. The gap atσ=0.1 is nearly eight-fold between ER-KAN and ChebyKAN. The mechanism is the derivative bound: degree-8 Chebyshev polynomials can amplify input perturbations by up tod 2 = 64, while our Gaussian RBFs are bounded by approximately 3.7 (Section 3). Figure 3 visualises the per-function degradation heatmap. 5.3 Absolute Accuracy Under Noise Although ER-KAN’s absolute RMSE underσ=0.1 is higher than ChebyKAN’s (0.158 vs 0.083) because ChebyKAN starts from a much lower clean baseline (0.025 vs 0.132), the key practical implication is about predictability: ER-KAN’s performance changes by only 20% between clean and noisy training (0.132→0.158), while ChebyKAN’s more than triples (0.025→0.083). A practitioner who cannot control the noise level in their measurement pipeline will find ER-KAN’s behaviour far easier to reason about. There are also functions where ER-KAN outperforms both polynomial-basis KANs in absolute terms under noise. On the 2D quadratic function (N= 50,σ= 0.1), ER-KAN achieves RMSE 0.071 versus 0.240 for vanilla KAN and 0.172 for ChebyKAN (and 0.140 for MLP). On the 2D sinusoidal function, ER-KAN (0.265) substantially outperforms MLP (0.415). Multidimensional inputs appear to be a particular strength of the shared-basis design. 5.4 Sweep Across Sample Sizes Figure 5 traces geomean RMSE acrossN ∈50,200,500atσ=0.1. ER-KAN’s RMSE drops from 0.158 (N=50) to 0.076 (N=500), while ChebyKAN drops from 0.083 to 0.028. The relative gap narrows with data: at N=500, ChebyKAN is 2.7×better in RMSE but “only” 2.7×more fragile to noise (vs 7.5×at N=50). The degradation ratio advantage is thus most pronounced exactly in the data-scarce regime. 7 0.00.030.1 Noise σ abs gauss_cos quad2d runge sin2d sin_pi step xsin3x Function 1.941.911.54 0.991.001.00 0.280.310.31 2.813.321.63 0.460.410.41 10.9911.336.55 0.991.061.06 4.522.272.23 ER-KAN vs MLP 0.00.030.1 Noise σ abs gauss_cos quad2d runge sin2d sin_pi step xsin3x Function 47.6617.243.37 3032.44893.31181.56 0.200.270.23 85.9445.928.34 7.5610.514.99 66.1019.894.89 3.693.723.00 419.8862.8513.04 ER-KAN vs Vanilla KAN 0.00.030.1 Noise σ abs gauss_cos quad2d runge sin2d sin_pi step xsin3x Function 57.3818.343.30 847.12379.57122.30 1.560.960.38 232.3460.448.58 24.7915.795.28 68.1719.954.86 4.113.952.95 667.3177.0012.81 ER-KAN vs ChebyKAN 2 4 6 8 10 MSE ratio (ER-KAN / comparator) 500 1000 1500 2000 2500 3000 MSE ratio (ER-KAN / comparator) 100 200 300 400 500 600 700 800 MSE ratio (ER-KAN / comparator) ER-KAN clean-test MSE ratio vs each comparator (geomean across N) Green < 1.0 = ER-KAN better | Red > 1.0 = comparator better Figure 3: Per-function ER-KAN MSE ratio vs each comparator (three panels: vs MLP, vs vanilla KAN, vs ChebyKAN), geomean acrossN∈50,200,500. Green (<1) means ER-KAN is better; red (>1) means the comparator is better. ER-KAN consistently beats ChebyKAN and vanilla KAN as noise grows (right two panels), while trailing MLP on most clean and low-noise 1D functions (left panel). σ=0.0σ=0.03σ=0.1 0.00 0.02 0.04 0.06 0.08 0.10 Geomean clean-test RMSE (all N) 0.097 0.102 0.113 0.078 0.084 0.102 0.017 0.024 0.047 0.013 0.022 0.047 Accuracy vs noise level ER-KAN MLP Vanilla KAN ChebyKAN ER-KAN MLP Vanilla KAN ChebyKAN 0 2 4 6 8 10 Noise degradation ratio (MSE at σ =0.1 / MSE at σ =0) 1.42× 1.66× 7.88× 10.61× Noise robustness (N=50, lower = more robust) No degradation (ratio=1) Analytic function suite — aggregate metrics across 8 functions and all N Figure 4: Summary across all 8 functions and allN: geomean RMSE per noise level (left) and noise degradation ratio atσ=0.1 (right), all four models. ChebyKAN wins on RMSE but pays a 10.6× degradation penalty; ER-KAN has the lowest degradation ratio of any model tested. Table 3 gives the full 25-cell (5 sample counts×5 noise levels) sweep from the scarcity/noise benchmark, showing ER-KAN is non-inferior to MLP in 5 out of 25 cells and actually superior in 2 cells under the 25%-margin criterion. 5.5 External Baseline Comparison ER-KAN achieves geomean MSE ratio 1.01 versus MLP—essentially equivalent accuracy at 0.61× the training speed (Table 4). Against the officialefficient-kan[Blealtan, 2024], ER-KAN’s MSE is 1.44×higher but training is 1.97×faster, reflecting a speed-accuracy trade-off that practitioners can evaluate for their use case. 8 50200500 Train N 10 −2 10 −1 Geomean RMSE σ = 0.0 50200500 Train N 10 −1 2 × 10 −2 3 × 10 −2 4 × 10 −2 6 × 10 −2 Geomean RMSE σ = 0.03 50200500 Train N 10 −1 3 × 10 −2 4 × 10 −2 6 × 10 −2 Geomean RMSE σ = 0.1 Geomean clean-test RMSE across 8 functions vs training samples ER-KANMLPVanilla KANChebyKAN Figure 5: Geomean clean-test RMSE across 8 functions vs training samples, for all four models at each noise level (σ∈0,0.03,0.1). ChebyKAN and vanilla KAN lead atσ=0 but their curves rise steeply with noise; ER-KAN and MLP show flat trajectories, with ER-KAN’s the flattest of all. 2550 100250500 3 × 10 1 4 × 10 1 6 × 10 1 2 × 10 2 3 × 10 2 4 × 10 2 Training samples 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 Clean MSE (mean ± SD) σ = 0.0 2550 100250500 3 × 10 1 4 × 10 1 6 × 10 1 2 × 10 2 3 × 10 2 4 × 10 2 Training samples 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 σ = 0.02 2550 100250500 3 × 10 1 4 × 10 1 6 × 10 1 2 × 10 2 3 × 10 2 4 × 10 2 Training samples 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 σ = 0.06 2550 100250500 3 × 10 1 4 × 10 1 6 × 10 1 2 × 10 2 3 × 10 2 4 × 10 2 Training samples 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 σ = 0.2 Sample efficiency: clean MSE vs training size at selected noise levels MLPVanilla KANER-KAN Figure 6: Sample efficiency on the single-oscillator regression task. 5.6 Timing and Compute-Tuned Comparison Figure 10 and Table 5 break down warm-up, training, and inference. ER-KAN’s 2.7×training speedup over vanilla KAN—and the even larger 4.6×inference speedup—makes it practical for settings where vanilla KAN is too slow to tune or deploy. Even when MLP is given the same compute budget as ER-KAN (a 66,561-parameter MLP whose training time matches ER-KAN’s within 5%), ER-KAN matches or beats it in 3 of the 5 tested regimes (Table 6). 5.7 Ablation Study Table 7 and Figure 13 isolate component contributions. Basis sharing is the most important: removing it (reverting to per-edge RBFs as in FastKAN) increases geomean MSE by 54% and causes statistically significant degradation in 4 of 5 regimes. Curriculum noise injection contributes a 7% geomean improvement. Adaptive regularisation has no measurable effect in our experiments—we include it for robustness but do not claim it as a contributor. 5.8 PINN: Damped Harmonic Oscillator On the oscillator PINN, ER-KAN achieves best-solution MSE 1.17×10 −7 —a 4.2×improvement over MLP’s 4.91×10 −7 (Table 8). Vanilla KAN is intermediate at 7.99×10 −6 (surprisingly, 9 Table 3: Full scarcity/noise sweep headline (25 cells = 5 sample counts× 5 noise levels, 10 paired seeds, 25% non-inferiority margin, 95% bootstrap CI). Comparator Cells Non-inf. Acc. sup. Acc. inf. ≥3× speedup ER-KAN / Comp. med. time MLP2552100.51 / 0.36 s Vanilla KAN 2500520.51 / 1.23 s 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 5.98e-026.50e-046.88e-053.70e-053.10e-05 3.97e-021.60e-033.95e-041.47e-046.89e-05 6.03e-022.31e-022.33e-038.00e-043.30e-04 7.43e-021.28e-025.16e-032.10e-038.23e-04 1.13e-013.22e-021.58e-026.73e-032.93e-03 MLP 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 3.62e-021.07e-034.39e-051.01e-053.96e-06 3.31e-022.25e-032.82e-049.83e-054.12e-05 4.63e-025.80e-031.61e-034.95e-043.28e-04 6.18e-021.77e-023.95e-031.21e-037.58e-04 1.09e-016.71e-022.28e-024.09e-032.26e-03 Vanilla KAN 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 5.45e-021.12e-031.66e-044.97e-063.46e-06 4.15e-022.90e-034.73e-041.25e-046.67e-05 4.71e-028.25e-032.52e-036.41e-043.59e-04 5.88e-022.17e-026.30e-031.79e-038.80e-04 9.84e-026.76e-022.47e-026.17e-032.69e-03 ER-KAN 10 −4 10 −3 10 −2 10 −1 Clean MSE 10 −5 10 −4 10 −3 10 −2 10 −1 Clean MSE 10 −5 10 −4 10 −3 10 −2 Clean MSE Mean squared error across scarcity / noise grid (10 seeds per cell) Figure 7: MSE heatmaps across all 25 cells (5 sample counts×5 noise levels) from the single- oscillator scarcity/noise benchmark (3 models; ChebyKAN was evaluated separately on the analytic function suite—see Figure 5). Each cell shows geomean MSE; darker is better. ER-KAN and MLP maintain relatively uniform colour across the noise axis; vanilla KAN lightens sharply (MSE rising) as noise increases. somewhat worse than both), with substantially longer training time. All models reached the residual target of 10 −4 on all seeds, so the differences in solution MSE reflect genuine accuracy differences rather than convergence failures. The Gaussian RBF basis is well-matched to the damped sinusoidal solution: Gaussians are universal approximators for smooth functions [Broomhead and Lowe, 1988], and the localised basis can represent the amplitude decay across the time domain without the Gibbs-like oscillations that pure polynomial bases can produce. 5.9 PINN: Burgers’ Equation None of the models converge on Burgers’ withν=0.01/π. MLP is the least bad at 10.3% mean L2 relative error; the KAN variants reach 19.5–26.7%. We discuss why in Section 6; the short version is that Adam with 20,000 epochs is insufficient for this problem, and smooth basis functions are Table 4: Full scarcity/noise grid with calibratedefficient-kanbaseline (25 cells, 10 paired seeds, 25% non-inferiority margin). MSE ratio and speedup are geomeans across all 25 cells. ComparatorCells Non-inf. Acc. sup. Acc. inf. ≥3× spd MSE ratio Speedup MLP2552101.010.61× Vanilla KAN2500521.481.96× efficient-kan 2522601.441.97× 10 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 1.801.671.050.140.14 2.151.731.230.941.07 1.660.740.920.881.12 1.101.521.070.961.07 0.992.021.220.950.93 MSE ratio vs MLP 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 3.624.411.710.830.91 3.151.541.681.421.63 1.771.341.291.241.11 1.341.371.351.461.15 1.041.181.101.481.27 MSE ratio vs Vanilla KAN 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 0.600.580.500.340.32 0.660.920.920.700.62 0.670.670.660.510.69 0.600.670.650.580.62 0.550.640.600.570.68 Speed ratio vs MLP 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 1.711.502.032.252.54 1.421.562.313.674.43 1.331.311.662.373.20 1.371.231.592.403.19 1.281.561.662.142.48 Speed ratio vs Vanilla KAN 0.5 1.0 1.5 2.0 ER-KAN / comparator MSE 0.4 0.5 0.6 0.7 0.8 0.9 1.0 comparator / ER-KAN time 1 2 3 4 ER-KAN / comparator MSE 1.0 1.5 2.0 2.5 3.0 3.5 4.0 comparator / ER-KAN time ER-KAN relative accuracy and speed (paired geometric-mean ratio, 10 seeds) Figure 8: Relative MSE heatmaps from the single-oscillator benchmark: each cell shows the ratio of ER-KAN MSE to the comparator’s MSE (MLP left, vanilla KAN right). Values<1 (blue) mean ER-KAN is better; values>1 (red) mean the comparator is better. ER-KAN gains relative to vanilla KAN as noise increases; it trails MLP except in the high-noise, low-Ncorner. (ER-KAN vs ChebyKAN ratios appear in Figure 3.) architecturally mismatched to a near-discontinuous shock. 6 Discussion When does the noise degradation ratio matter? TheD R measures proportional sensitivity, not absolute error. ChebyKAN still achieves lower absolute RMSE than ER-KAN atσ= 0.1 because it started from a much lower clean-data baseline. TheD R becomes the decisive metric in two scenarios: (1) when you want to deploy the same model architecture across varying noise conditions—ER-KAN’s flat trajectory makes it easier to set expectations; (2) when noise is higher thanσ=0.1, where ER-KAN’s 1.4×compounding eventually crosses ChebyKAN’s 10.6×. For purely low-noise or clean-data applications, ChebyKAN is the better choice. Why is basis sharing the most important component? Intuitively: per-edge basis parameters allow each edge to fit noise independently, leading to edge-by-edge overfitting. Shared centres force all edges to explain the data with a common latent representation, acting as a regulariser that noise augmentation alone cannot replicate. This is confirmed by the ablation (Table 7): even without 11 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 1.801.671.050.140.14 2.151.731.230.941.07 1.660.740.920.881.12 1.101.521.070.961.07 0.992.021.220.950.93 MSE ratio vs MLP 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 3.624.411.710.830.91 3.151.541.681.421.63 1.771.341.291.241.11 1.341.371.351.461.15 1.041.181.101.481.27 MSE ratio vs Vanilla KAN 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 5.854.681.320.300.32 3.281.851.771.721.50 2.151.011.281.351.30 1.691.251.341.421.34 1.101.721.401.281.07 MSE ratio vs Official eff-KAN (calibrated, ±2 grid) 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 0.600.580.500.340.32 0.660.920.920.700.62 0.670.670.660.510.69 0.600.670.650.580.62 0.550.640.600.570.68 Speed ratio vs MLP 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 1.711.502.032.252.54 1.421.562.313.674.43 1.331.311.662.373.20 1.371.231.592.403.19 1.281.561.662.142.48 Speed ratio vs Vanilla KAN 2550100250500 Training samples 0.0 0.02 0.06 0.1 0.2 Noise σ 3.301.721.361.812.23 1.712.021.974.143.76 1.321.761.432.523.64 1.341.241.262.883.72 1.241.171.222.132.89 Speed ratio vs Official eff-KAN (calibrated, ±2 grid) 0.5 1.0 1.5 2.0 ER / comparator 0.4 0.5 0.6 0.7 0.8 0.9 1.0 comparator / ER time 1 2 3 4 ER / comparator 1.0 1.5 2.0 2.5 3.0 3.5 4.0 comparator / ER time 1 2 3 4 5 ER / comparator 1.0 1.5 2.0 2.5 3.0 3.5 4.0 comparator / ER time ER-KAN vs baselines — full scarcity/noise grid with calibrated official efficient-kan (paired geometric-mean ratio, 10 seeds) Figure 9: Full scarcity/noise grid including the officialefficient-kanlibrary. ER-KAN is non- inferior to MLP in 5 cells (geomean ratio 1.01) and competitive withefficient-kan(ratio 1.44, but 1.97× faster). Table 5: Training phase timing on CPU (500 observations, 5 seeds, mean±SD). Warm-up = first epoch. Training = remaining epochs. Inference latency measured over 20 repetitions on the full test set. ModelParams Warm-up (ms) Training (s) Inference (ms) Inf. lat. (μs/samp.) MLP4,353 16.0 ± 33.6 0.144 ± 0.106 0.080 ± 0.0321.07 ± 0.43 ER-KAN5952.2 ± 1.50.296 ± 0.016 0.132 ± 0.0091.76 ± 0.12 Vanilla KAN8017.7 ± 9.90.799 ± 0.034 0.602 ± 0.0288.02 ± 0.37 curriculum noise (−7%), the shared basis provides most of the robustness (+54% when removed). Why does ER-KAN excel on 2D functions? The quadratic and sinusoidal 2D functions require the network to model interaction terms betweenx 1 andx 2 . MLP handles this via the nonlinear activation; KAN-family models handle it through the composition of univariate functions. The shared Gaussian basis appears to provide a better inductive bias for smooth multivariate composition than either Chebyshev or B-spline bases under noise, perhaps because the localised RBF activations reduce cross-term interference. This warrants further theoretical investigation. Why does ER-KAN win on the oscillator PINN? Physics-informed training removes labeled data entirely; the network must learn purely from the ODE residual. The Gaussian RBF basis provides several advantages here: (a) Gaussian functions are naturally well-suited to modelling exponentially decaying oscillations (the exact solution ise −ζωt cos(ω d t) form); (b) shared bases prevent the network from finding degenerate residual-minimising solutions that generalise poorly; and (c) the ER-KAN parameter count (595) is well-matched to the problem’s degrees of freedom. 12 MLP ER-KAN Vanilla KAN −20 −10 0 10 20 30 40 50 First-epoch warm-up (ms) 16.05 2.19 7.67 MLP ER-KAN Vanilla KAN 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Training time — excl. warm-up (s) 0.14 0.30 0.80 MLP ER-KAN Vanilla KAN 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Inference — full test set (ms) 0.08 0.13 0.60 Timing breakdown: warm-up · training · inference (mean ± SD, 5 seeds, CPU, 500 training observations) Figure 10: Training-phase timing breakdown. ER-KAN is 2.7×faster than vanilla KAN per epoch (0.296 vs 0.799 s) and has 4.6× lower inference latency. 020040060080010001200 Epoch 10 −1 10 0 Normalized MSE Seed 7 020040060080010001200 Epoch 10 −1 10 0 Seed 41 020040060080010001200 Epoch 10 −1 10 0 Seed 73 Convergence curves — CPU (full-resolution normalized MSE, seeds 7/41/73) MLPVanilla KANER-KANTrainVal Figure 11: Training loss convergence curves (median over 5 seeds). ER-KAN and MLP reach low residuals at similar epoch counts; vanilla KAN is slower to converge and more variable across seeds. Why does all of this fail on Burgers’? The canonical failure mode [Krishnapriyan et al., 2021] for Burgers’ with smallνis that the shock layer att≈0.8 concentrates the PDE residual in a small spatial region that first-order optimisers cannot find without adaptive sampling or second-order methods. MLP’s piecewise-smooth representational capacity gives it a minor advantage over smooth KAN bases—a smooth function cannot approximate a near-discontinuity without Gibbs-like ringing. The correct approach for this problem is L-BFGS with adaptive collocation [Lu et al., 2021] or causal loss weighting [Wang et al., 2024]; no architecture in our comparison addresses this at the training-protocol level. Recommendation. We suggest future efficient-KAN papers reportD R alongside RMSE. Com- puting it requires only one additional condition (repeat the evaluation with noisy training data) and reveals robustness properties that RMSE on clean data completely hides. 13 N=25 σ=0.2 N=50 σ=0.02 N=100 σ=0.06 N=250 σ=0.02 N=500 σ=0.06 10 −4 10 −3 10 −2 10 −1 Clean MSE (mean ± SD) Accuracy (ER-KAN: 595 | MLP-std: 4,353 | MLP-tuned: 66,561 params) ER-KAN MLP MLP (compute-tuned) N=25 σ=0.2 N=50 σ=0.02 N=100 σ=0.06 N=250 σ=0.02 N=500 σ=0.06 0.0 0.2 0.4 0.6 0.8 Training time (s, mean ± SD) ER-KAN MLP MLP (compute-tuned) Compute-tuned MLP: equal wall-clock budget per training regime (MLP-tuned uses widest hidden layer that fits within ER-KAN's training time, 5 seeds) Figure 12: Compute-tuned MLP: the widest MLP (66,561 params) whose training time matches ER-KAN’s at each N. ER-KAN is competitive with this much larger model in 3 of 5 regimes. Full ER-KAN –Curriculum noise –Adaptive reg. –Residual path –Shared bases Variant N=25,σ=0.2 N=50,σ=0.02 N=100,σ=0.06 N=250,σ=0.02 N=500,σ=0.06 Regime 1.15e-01 * 6.22e-03 * 3.38e-03 * 2.78e-04 * 5.60e-04 * 9.84e-029.84e-028.19e-029.45e-02 2.90e-034.22e-032.90e-032.75e-03 2.52e-033.15e-032.52e-032.19e-03 1.25e-041.19e-041.25e-041.49e-04 3.59e-043.45e-043.59e-043.68e-04 Ablation: mean MSE per variant and regime 10 −3 10 −2 10 −1 Mean clean MSE Figure 13: Ablation heatmap: geomean MSE ratio (vs full ER-KAN) across 5 noise–Nregimes. Basis sharing is the dominant factor; adaptive regularisation has no measurable effect. 7 Limitations and Future Work Clean-data performance. ER-KAN does not match polynomial-basis KANs when data is clean; any deployment where training and test conditions are both low-noise should prefer ChebyKAN or vanilla KAN. Absolute accuracy under noise. Even atσ=0.1, ChebyKAN’s absolute RMSE is lower than ER-KAN’s for most 1D functions. The crossover (where ER-KAN’s stability advantage dominates in absolute terms) occurs at noise levels above those tested here. Characterising this crossover more precisely is left to future work. Shock-dominated PDEs. All models fail on Burgers’ withν=0.01/π. Coupling ER-KAN with adaptive collocation or L-BFGS is an open direction. Adaptive RBF centres. We use fixed uniformly spaced centres. Adaptive centre placement— concentrating centres where the function varies rapidly—could improve clean-data performance without sacrificing noise robustness. 14 Table 6: Compute-tuned MLP baseline (5 seeds, mean±SD). MLP-tuned is the widest hidden-layer MLP whose training time matches ER-KAN’s within 5%. Hidden width 256 (66,561 params) satisfied the budget in all regimes. RegimeModelParamsClean MSETime (s) N = 25, σ = 0.2ER-KAN595 1.051e− 01± 9.0e− 02 0.300 MLP4,353 1.066e− 01± 6.6e− 02 0.181 MLP (compute-tuned) 66,561 1.029e− 01± 5.7e− 02 0.300 N = 50, σ = 0.02ER-KAN595 3.713e− 03± 2.1e− 03 0.439 MLP4,353 3.568e− 03± 4.4e− 03 0.448 MLP (compute-tuned) 66,561 4.301e− 03± 3.9e− 03 0.439 N = 100, σ = 0.06ER-KAN595 2.303e− 03± 6.1e− 04 0.346 MLP4,353 2.762e− 03± 1.2e− 03 0.253 MLP (compute-tuned) 66,561 2.711e− 03± 1.3e− 03 0.346 N = 250, σ = 0.02ER-KAN595 1.308e− 04± 3.8e− 05 0.739 MLP4,353 1.030e− 04± 1.3e− 05 0.525 MLP (compute-tuned) 66,561 1.337e− 04± 6.3e− 05 0.739 N = 500, σ = 0.06ER-KAN595 3.359e− 04± 4.7e− 05 0.612 MLP4,353 3.191e− 04± 8.0e− 05 0.522 MLP (compute-tuned) 66,561 3.407e− 04± 8.0e− 05 0.613 Table 7: Module ablation results (10 paired seeds, 5 regimes, geomean MSE ratio vs. complete ER-KAN; ratio >1 means worse than full). VariantParams Geomean MSE ratio Cells sign. improved Cells sign. harmed Complete ER-KAN5951.0000 −Adaptive reg.5951.0000 −Residual path5311.0300 −Curriculum noise5951.0701 −Shared bases (unshared RBF)1,1371.5404 Dimensionality. Our highest-dimensional experiment is 2D. In higher dimensions (d >5), the shared-basis approach may need modification to avoid the curse of dimensionality. Adaptive regularisation. The entropy-weighted penalty had no measurable effect. Under- standing when (if ever) it contributes, and whether a different regularisation design would help, is an open question. 8 Conclusion We introduced ER-KAN, a 595-parameter Kolmogorov–Arnold Network variant with shared Gaussian RBF bases, curriculum noise injection, and adaptive regularisation. Its core finding is a noise degradation ratio of 1.4×—compared with 7.9×for vanilla KAN and 10.6×for ChebyKAN— measured across eight analytic functions with σ=0.1 noise at N =50. This work does not claim ER-KAN is the best efficient KAN in all settings. On clean 15 –Curriculum noise –Adaptive reg. –Residual path –Shared bases Removed / changed module 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 MSE ratio vs full ER-KAN (>1 = worse) Accuracy impact of ablating each module Regime N=25,σ=0.2 N=50,σ=0.02 N=100,σ=0.06 N=250,σ=0.02 N=500,σ=0.06 –Curriculum noise –Adaptive reg. –Residual path –Shared bases Removed / changed module 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Time ratio vs full ER-KAN (<1 = faster) Training-time impact of ablating each module Figure 14: Ablation: geomean MSE ratio vs full ER-KAN at each noise–Nregime. Each bar above 1.0 indicates how much that component contributes. Basis sharing (orange) consistently dominates; curriculum noise (green) provides a smaller but consistent gain; adaptive regularisation (blue) is flat. Table 8: Physics-informed damped oscillator (ζ= 0.15,ω= 2.0) solved without observed data (5 seeds, 5000 epochs, CPU). All models reached residual target<10 −4 on all seeds. T = time to first breach target (seconds). ModelParams Best sol. MSE Residual MSE Time (s) T (s) MLP4,353 4.91e− 073.43e− 054.5± 0.32.21 ER-KAN595 1.17e− 071.39e− 05 41.8± 1.13.69 Vanilla KAN801 7.99e− 061.19e− 05 75.2± 6.25.55 data, ChebyKAN is the better approximator, and we say so clearly. What ER-KAN offers is stable, predictable behaviour as noise grows—a property that matters in the physical sciences and engineering, where measurement noise is the rule, not the exception. The noise degradation ratio is a simple, one-line addition to any function-approximation evalua- tion that reveals robustness properties currently invisible in the literature. We hope it becomes standard practice. Table 9: Burgers’ PINN (ν=0.01/π): L2 relative error (mean±std, 5 seeds). No model reached the <1% target. MLP is the least bad. ModelParams L2 rel. error Reached <1% MLP8,577 0.103± 0.0520/5 ER-KAN808 0.195± 0.0500/5 Vanilla KAN504 0.255± 0.0490/5 ChebyKAN600 0.267± 0.0210/5 16 MLP ER-KAN Vanilla KAN 10 −7 10 −6 10 −5 Best solution MSE 4.9e-07 1.2e-07 8.0e-06 MLP ER-KAN Vanilla KAN 10 −5 6 × 10 −6 2 × 10 −5 3 × 10 −5 4 × 10 −5 Final ODE residual MSE 3.4e-05 1.4e-05 1.2e-05 MLP ER-KAN Vanilla KAN 0 10 20 30 40 50 60 70 80 Total training time (s) 4.5e+00 4.2e+01 7.5e+01 T=2.2s (5/5 seeds) T=3.7s (5/5 seeds) T=5.6s (5/5 seeds) Physics-informed neural network — damped oscillator (ζ=0.15, ω=2.0) (residual loss only, 200 collocation points, 5000 epochs, 5 seeds, CPU) Figure 15: PINN oscillator: predicted solution and residual for each model. ER-KAN tracks the exact solution most accurately. Acknowledgments This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Broader impacts. This work studies architectural choices for scientific machine learning under noisy, data-scarce conditions. We do not foresee direct negative societal impacts from the method itself. Improved surrogate modelling for expensive simulations could reduce energy use in materials discovery and computational fluid dynamics. Practitioners should validate independently before deploying in safety-critical settings, as our evaluation is limited to smooth analytic functions and one ODE problem. Author contributions. All experimental design, model implementation, training runs, and scientific interpretation were performed by the authors. 17 References Blealtan. efficient-kan: Efficient KAN implementation.https://github.com/Blealtan/ efficient-kan, 2024. Zavareh Bozorgasl and Hao Chen. WaveKAN: Wavelet Kolmogorov–Arnold networks. arXiv preprint arXiv:2405.12832, 2024. David S Broomhead and David Lowe. Multivariable functional interpolation and adaptive networks. 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Integrating scientific knowledge with machine learning for engineering and environmental systems. ACM Computing Surveys, 55(4):1–37, 2022. 19 Table 10: Single-oscillator benchmark results (500 observations, 5 seeds, mean± SD). CPU: Apple M3 Pro; MPS: Apple M3 Pro GPU. ModelParams CPU Clean MSE CPU Time (s) MPS Clean MSE MPS Time (s) MLP4,353 4.52e-04± 7.7e-05 0.70± 0.43 4.76e-04± 1.1e-04 3.72± 1.92 Vanilla KAN801 3.38e-04± 3.9e-05 4.38± 1.51 3.25e-04± 6.2e-05 10.96± 1.92 ER-KAN675 4.79e-04± 1.7e-04 0.66± 0.09 4.36e-04± 1.7e-04 3.85± 1.06 Table 11: ODE surrogate results (800 fits: 4 trajectory counts× 4 noise levels× 10 seeds× 5 models). Non-inferior: ER-KAN MSE within 25% of comparator. ModelParams Median cell MSE Median time (s) Non-inf. cells MLP4,4810.00650.750 Parameter-matched MLP1,1480.00600.610 Vanilla KAN1,5690.03713.018 ER-KAN1,1710.03890.70— Official efficient-kan1,6640.03461.995 A Full Scarcity–Noise Sweep B ODE Surrogate Results 5102050 6 × 10 0 3 × 10 1 4 × 10 1 Training trajectories 10 −3 10 −2 10 −1 MSE (mean ± SD) σ = 0.0 5102050 6 × 10 0 3 × 10 1 4 × 10 1 Training trajectories 10 −3 10 −2 10 −1 σ = 0.03 5102050 6 × 10 0 3 × 10 1 4 × 10 1 Training trajectories 10 −3 10 −2 10 −1 σ = 0.06 5102050 6 × 10 0 3 × 10 1 4 × 10 1 Training trajectories 10 −2 10 −1 σ = 0.1 ODE surrogate: MSE vs training trajectories at selected noise levels MLPParameter-matched MLPVanilla KANER-KANOfficial efficient-kan Figure 16: ODE surrogate sample efficiency curves. C Additional Figures D Reproducibility Hardware. All CPU experiments ran on an Apple M3 Pro (macOS). No GPU was used for the main experiments (the benchmark table reports MPS timings for completeness, but all comparisons use CPU). Software. PyTorch 2.x, SciPy (reference solutions), NumPy. No external KAN library was used for ER-KAN, ChebyKAN, or vanilla KAN implementations; they are written from scratch 20 5102050 6 × 10 0 3 × 10 1 4 × 10 1 Training trajectories 1 2 3 4 5 Median training time (s) MLP Parameter-matched MLP Vanilla KAN ER-KAN Official efficient-kan 5102050 6 × 10 0 3 × 10 1 4 × 10 1 Training trajectories 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Speed ratio (comparator / ER-KAN) MLP Parameter-matched MLP Vanilla KAN Official efficient-kan ODE surrogate training time (median across noise levels, CPU) Figure 17: ODE surrogate timing breakdown. 5102050 Train traj. 0.0 0.03 0.06 0.1 Noise σ 2.859.5923.018.00 2.676.7212.817.21 2.084.736.166.05 1.652.863.744.43 MSE ratio vs MLP 5102050 Train traj. 0.0 0.03 0.06 0.1 Noise σ 3.079.9120.987.16 2.387.2112.896.09 2.005.266.906.46 1.813.664.445.21 MSE ratio vs Param-matched MLP 5102050 Train traj. 0.0 0.03 0.06 0.1 Noise σ 0.991.021.261.22 0.950.971.171.16 0.891.011.101.18 0.861.071.241.38 MSE ratio vs Vanilla KAN 5102050 Train traj. 0.0 0.03 0.06 0.1 Noise σ 1.181.031.725.62 1.150.961.493.01 1.071.021.372.14 1.001.071.381.98 MSE ratio vs Official eff-KAN 5102050 Train traj. 0.0 0.03 0.06 0.1 Noise σ 0.881.181.230.48 0.690.971.340.66 0.540.971.100.98 0.620.620.951.01 Speed ratio vs MLP 5102050 Train traj. 0.0 0.03 0.06 0.1 Noise σ 0.621.020.980.37 0.490.721.050.50 0.370.710.990.70 0.470.570.810.79 Speed ratio vs Param-matched MLP 5102050 Train traj. 0.0 0.03 0.06 0.1 Noise σ 7.605.084.194.31 4.203.443.692.99 2.903.473.413.09 3.553.103.223.28 Speed ratio vs Vanilla KAN 5102050 Train traj. 0.0 0.03 0.06 0.1 Noise σ 3.513.253.093.52 2.652.493.413.73 2.052.773.023.08 2.882.403.003.05 Speed ratio vs Official eff-KAN 5 10 15 20 ER / comparator 0.6 0.8 1.0 1.2 comparator / ER time 5 10 15 20 ER / comparator 0.4 0.6 0.8 1.0 comparator / ER time 0.9 1.0 1.1 1.2 1.3 ER / comparator 2 4 6 comparator / ER time 1 2 3 4 5 ER / comparator 1.0 1.5 2.0 2.5 3.0 3.5 comparator / ER time ODE surrogate — ER-KAN paired comparisons (geometric-mean ratio, 10 seeds) Figure 18: ODE surrogate MSE heatmaps. 2550 100250500 3 × 10 1 4 × 10 1 6 × 10 1 2 × 10 2 3 × 10 2 4 × 10 2 Training samples 0.5 1.0 1.5 2.0 2.5 Median training time (s) MLP Vanilla KAN ER-KAN 2550 100250500 3 × 10 1 4 × 10 1 6 × 10 1 2 × 10 2 3 × 10 2 4 × 10 2 Training samples 0.5 1.0 1.5 2.0 2.5 3.0 Speed ratio (comparator / ER-KAN) Vanilla / ER-KAN MLP / ER-KAN Training-time comparison (median across noise levels, CPU) Figure 19: Training time distribution across architectures (box plots, 5 seeds). 21 10 0 10 1 10 2 10 3 Batch size 10 −1 10 0 10 1 10 2 Latency (μs / sample) Per-sample inference latency vs batch size (μs / sample, 5 seeds, CPU) MLP ER-KAN Vanilla KAN Figure 20: Inference latency per sample (μs): detailed breakdown with confidence intervals. in the experiment scripts. Seeds. All experiments use 5 seeds (0–4) for weight initialisation and data sampling. Seeds are set globally before each run via torch.manual_seed and np.random.seed. Hyperparameters. ER-KAN:G= 8, hidden dim 32, 2 layers,σ base = 0.1,λ= 10 −4 , Adam LR 10 −3 →10 −5 (cosine), batch 64, early stop patience 500. ChebyKAN: degree 8, hidden dim 32, 2 layers, zero-init residual. MLP: hidden dim 64, 2 layers (3 for PINN), Tanh. Vanilla KAN:G= 5 B-spline knots, hidden dim 32. PINN loss weights: oscillator 1:10:5, Burgers’ 1:20:20. Code. All scripts are included in the supplementary material. 22