PIKAN: Kolmogorov-Arnold Inspired Network

• PIKAN is a neural network architecture based on the Kolmogorov-Arnold theorem, employing edge-wise learnable univariate functions for adaptable, transparent function approximation in physics-informed scenarios.
• It integrates physical inductive biases by allowing fixed and hybrid activation functions that mirror known analytic forms, enabling direct symbolic extraction and interpretability.
• By harnessing expressive local adaptivity, PIKAN achieves significant parameter efficiency and competitive performance across tasks like UAV channel modeling, PDE solving, and inverse problems.

Physics-Inspired Kolmogorov-Arnold Network (PIKAN) is an advanced neural network architecture rooted in the Kolmogorov-Arnold representation theorem. PIKAN is specifically engineered to simultaneously achieve high accuracy, strong inductive bias from physical laws, and model interpretability, with significant parameter efficiency compared to conventional neural models. Its structure replaces fixed-weighted, node-activated multilayer perceptrons (MLPs) with edge-wise learnable univariate nonlinearities—commonly parameterized as B-splines, Chebyshev polynomials, wavelets, or other basis sets—suited for transparent function approximation in data-driven and physics-informed tasks.

1. Mathematical Foundations: Kolmogorov-Arnold Representation and Edge-wise Nonlinearity

At the core of PIKAN lies the Kolmogorov-Arnold representation theorem, which ensures that a continuous function $f: [0,1]^n \to \mathbb{R}$ can be exactly represented as a summation and nesting of univariate functions:

$$f(x_1, \ldots, x_n) = \sum_{q=1}^{2n+1} \Phi_q\left( \sum_{p=1}^n \phi_{q,p}(x_p) \right)$$

where all $\Phi_q$ and $\phi_{q,p}$ are continuous univariate functions. PIKAN operationalizes this result by replacing the weights of a traditional MLP with learnable univariate functions along edges, typically represented as either B-splines, Chebyshev polynomials, wavelets, Sinc, or Jacobi polynomials, depending on the application (Liu et al., 30 Apr 2024, Yu et al., 5 Oct 2024, Guo et al., 7 Nov 2024, Shuai et al., 13 Aug 2024, Gong et al., 23 Aug 2025, Jacob et al., 9 Nov 2024).

The forward pass in a PIKAN is given by, for each layer $\ell$:

$$x_{\ell+1, j} = \sum_{i=1}^{n_\ell} \phi_{\ell, j, i}(x_{\ell,i})$$

where $\phi_{\ell, j, i}$ is a learnable univariate function for each edge, providing both adaptive expressivity and local nonlinearity. This formulation generalizes to deeper and wider networks beyond the width prescribed by the theorem, improving smoothness and the adaptation of favorable scientific representations (Liu et al., 30 Apr 2024).

2. Physical Inductive Bias and Symbolic Interpretability

A distinguishing feature of PIKAN is the integration of physical principles as flexible inductive biases at the architectural level, not merely as loss penalties. Rather than treating physics-inspired information strictly as hard constraints (as in conventional PINNs), PIKAN enables the “fixing” of specific edge functions to match the analytic form of known physical effects. For UAV air-to-ground channel modeling, example activations include squaring (to induce $(distance)^2$ terms for free-space path loss), logarithmic scaling, or higher power functions (for multi-path or two-ray reflection models) (Tekbıyık et al., 7 Oct 2025):

• Example: setting an activation to $(\cdot)^2$ in order to align a learned path-loss component with the free-space loss formula
• Symbolification step: after training/pruning, learned edge functions can be “snapped” to known analytic types, directly yielding interpretable, symbolic expressions.

This mechanism allows PIKAN to remain flexible and accurately fit empirical data, while steering the network toward representations that are physically plausible and easily validated by domain experts. Symbolification is especially facilitated by pruning minimal architectures and aligning splines with analytic function candidates (e.g., $\{\text{log}, (\cdot)^2, (\cdot)^{\pm 4}, \sin\}$), providing closed-form expressions for practical use (Tekbıyık et al., 7 Oct 2025).

3. Parameter Efficiency and Model Performance

PIKAN leverages the expressive local adaptivity of edge-based univariate nonlinearities, enabling highly parsimonious models with strong empirical performance. Experiments demonstrate:

• For UAV path loss modeling, PIKAN achieves similar or better mean absolute error and correlation metrics (e.g., $\text{MAE} \approx 3.07$ dB, $\rho \approx 0.75$) compared to MLPs, while using only 232 parameters—up to 37-fold reduction relative to MLP baselines (8,500+ parameters) (Tekbıyık et al., 7 Oct 2025).
• Fast convergence, smooth loss curves, and superior accuracy are observed in scientific regression, PDE solving, and system identification tasks, frequently outperforming MLPs despite much smaller model sizes (Liu et al., 30 Apr 2024, Patra et al., 25 Jul 2024, Shuai et al., 13 Aug 2024).

Reduction in parameter count originates from the combinatorial expressive power of learnable edge functions and the theorem’s leverage of compositional separation, while the interpretable form is preserved by explicit access to each function along each edge.

4. Workflow: Integration of Physical Priors and Symbolic Model Extraction

PIKAN’s model development and deployment workflow features:

1. Architecture Definition: Specify, for each edge, whether the activation function should be learnable (generic spline/polynomial), “fixed” to a physically motivated analytic type, or a hybrid.
2. Training: Fit the network to data using standard optimizers (SGD, Adam, L-BFGS), optionally augmenting the loss with measurement error and (if desired) soft physics constraints.
3. Pruning: Apply sparsification via $L_1$ and entropy penalties to reduce redundant edges; prune to yield a minimal architecture.
4. Symbolification: Replace trained activation functions with best-fit analytic candidates (e.g., log, power, sinusoidal, rational), enabling extraction of interpretable, physically plausible equations.
5. Validation: Assess model performance both through fit to held-out data and through alignment of the extracted symbolic formula with underlying physics.

This flexible process facilitates the incorporation of physics as inductive bias. If measurement data diverges moderately from theory, the network can deviate from the analytic form where necessary, yet remain interpretable (Tekbıyık et al., 7 Oct 2025).

5. Applications and Implications for Scientific Modeling

PIKAN has been successfully applied to a variety of scientific and engineering problems, including but not limited to:

• Air-to-Ground Channel Modeling for UAVs: Fitting A2G channel models that combine symbolically interpretable propagation laws with empirical accuracy for beyond-5G/6G scenarios. Models are lightweight and deployable in resource-constrained systems (Tekbıyık et al., 7 Oct 2025).
• Partial Differential Equation Solving: Physics-informed variants demonstrate strong performance for ODEs and PDEs, with both data-driven and data-free settings, and scalability to high-dimensional cases using domain decomposition and separable architectures (Liu et al., 30 Apr 2024, Howard et al., 28 Jun 2024, Guo et al., 7 Nov 2024, Jacob et al., 9 Nov 2024).
• Parameter Identification and Inverse Problems: Superior parameter efficiency and reliability in noisy data and multiscale physics, compared to MLP-based PINNs (Shuai et al., 13 Aug 2024, Yu et al., 5 Oct 2024, Gong et al., 23 Aug 2025).
• Symbolic Regression and Scientific Discovery: “AI for Science” workflows where trained PIKANs yield interpretable closed-form laws, supporting hypothesis testing and scientific communication (Liu et al., 30 Apr 2024).

The method’s unique combination of flexibility, efficiency, and symbolic transparency provides a systematic bridge between black-box learning and traditional analytic modeling across multiple domains.

6. Distinction from PINNs and Training Considerations

PIKAN differs fundamentally from standard physics-informed neural networks (PINNs) in how it leverages physical knowledge:

• Rather than incorporating physics strictly via regularization or loss penalization, PIKAN can directly “fix” parts of its architecture (edges) to represent key physical operations.
• This design enables a tunable continuum between pure data-driven and pure physics-based modeling. If priors are uncertain, PIKAN retains flexibility; if physical structure is well-known, weight fixing guides learning toward those forms without hard constraints.

From a training perspective, PIKAN benefits from:

• Rapid convergence due to compact architectures and smooth loss landscapes (as a result of stable basis function choices and edge-wise adaptive nonlinearity).
• The possibility of explicit pruning and function replacement, further simplifying models for deployment and explainability.
• Scalability to larger or more complex domains, especially with parallelization, domain decomposition, or rank-separable variants for high-dimensional PDEs (Jacob et al., 9 Nov 2024).

7. Future Directions and Impact

PIKAN’s methodology—integrating learnable function composition and physics-inspired architectural bias—opens several avenues for future research:

• Broader adoption in settings demanding robust symbolic representations (e.g., interpretable operational modeling, standardization in wireless communications, system identification in engineering).
• Extension to larger, more complex domains through domain decomposition, separable networks for high-dimensional problems, and integration with operator-learning frameworks.
• Automated selection of physically plausible basis functions (“automatic symbolification”) and hybrid models to exploit both data-driven adaptivity and human insight.

A plausible implication is that PIKAN offers a template for next-generation scientific machine learning models that serve as both predictive engines and scientific collaborators, balancing expressive accuracy, symbolic interpretability, and rigorous integration of physical knowledge (Liu et al., 30 Apr 2024, Tekbıyık et al., 7 Oct 2025).

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Table 1. Key Features of PIKAN Compared to MLP/PINN

Feature	PIKAN	MLP/PINN
Edge-wise Activations	Learnable univariate functions; splines/polys	Fixed activations (ReLU, tanh)
Physical Prior Integration	As symbolically fixable edge activations	As loss term regularizer
Interpretability	Symbolic expressions easily extractable	Limited; post-hoc or black-box
Parameter Efficiency	Achieves high accuracy with few parameters	Requires large networks
Symbolification	Yes (direct architectural support)	No (black-box; rarely direct)

This architecture generalizes deep learning models by embedding functional decomposability and physical structure at the lowest architectural layer, enabling efficient, explainable, and theoretically grounded modeling in cutting-edge scientific and engineering domains.