PIKAN: Physics-Informed Kolmogorov–Arnold Network

• PIKAN is an energy-based solver that replaces MLPs with Kolmogorov–Arnold Networks to directly resolve multi-material elasticity problems.
• It employs B-spline activation functions and an admissible field construction to enforce exact boundary conditions and capture interface discontinuities.
• PIKAN achieves higher efficiency and accuracy compared to traditional PINNs, validated by numerical benchmarks on complex engineering problems.

A Physics-Informed Kolmogorov–Arnold Network (PIKAN) is an energy-based mesh-free solver for elasticity problems that replaces multilayer perceptrons (MLPs) with Kolmogorov–Arnold Networks (KANs) in the core of the Physics-Informed Neural Networks (PINNs) methodology. The primary innovation lies in leveraging the natural piecewise structure of KANs, particularly with B-spline activation functions, to directly and robustly resolve multi-material elasticity problems, including those with discontinuous material interfaces, without the need for explicit domain decomposition or interface penalty terms (Gong et al., 23 Aug 2025).

1. Variational Formulation and Physical Model

The governing equation is a linear elasticity problem over a multi-material body Ω. The total potential energy functional is given by

$$Π[u] = \int_{\Omega} \tfrac12 \varepsilon(u):C(x):\varepsilon(u)\, dΩ - \int_{\Omega} f \cdot u\, dΩ - \int_{Γ_{t}} \bar t \cdot u\, dΓ,$$

where:

• $$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$ is the small-strain tensor,
• $C(x)$ is the piecewise-constant fourth-order elasticity tensor distinguishing different materials,
• $f(x)$ is the prescribed body force,
• $\bar t$ is the prescribed traction on the Neumann boundary $Γ_t$.

PIKAN's loss functional is identical to the discretized total energy and is minimized over the class of admissible displacement fields.

2. Admissible Field Construction and KAN Ansatz

To enforce Dirichlet (essential) boundary conditions $u = \tilde{u}$ on $Γ_u$ exactly, PIKAN employs the admissible field ansatz

$u^{\rm pred}(x) = P(x) + D(x) \odot F(x;\theta),$

where:

• $P(x)$ is an extension of the boundary data into the interior,
• $$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$0 is the distance to the Dirichlet boundary ($$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$1 on $$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$2),
• $$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$3 is component-wise multiplication,
• $$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$4 is a KAN sub-network parameterized by $$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$5, constructed to vanish on $$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$6.

This guarantees that Dirichlet constraints are satisfied exactly without the need for penalty terms or constraint relaxation.

3. Kolmogorov–Arnold Network Architecture

KANs are based on the Kolmogorov–Arnold superposition theorem, which states that any continuous multivariate function $$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$7 can be decomposed as

$$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$8

with each $$\varepsilon(u) = \frac{1}{2} (\nabla u + \nabla u^\top)$$9 and $C(x)$0 being univariate functions. In PIKAN, each $C(x)$1 is implemented using trainable B-splines: $C(x)$2 where $C(x)$3 and $C(x)$4 are learnable weights, $C(x)$5 is a fixed activation (e.g., tanh), and each $C(x)$6 is a B-spline expansion. The final output layer is linear to match the possible unboundedness of physical displacements.

This architecture makes KANs inherently capable of representing piecewise-smooth solutions, offering a critical advantage for modeling fields with discontinuities in material properties or solution gradients.

4. Loss Function, Integration Schemes, and Automatic Differentiation

PIKAN employs the Deep Energy Method, minimizing the total potential energy: $C(x)$7 using quadrature for loss evaluation: $C(x)$8 where points $C(x)$9 and weights $f(x)$0 can be chosen as uniform grids, triangular centroids, Delaunay-based control areas, or Monte Carlo samples. Automatic differentiation is used to compute gradients of the loss with respect to all trainable parameters.

5. Interface Handling and Multi-Material Robustness

The use of B-spline activations in the KAN structure makes the parametric representation naturally suited for piecewise-smooth functions, meaning the model can represent displacement fields that are continuous but have abrupt changes in their derivatives (strains) at multi-material interfaces. There is no need for subdomain decomposition, interface-specific penalty terms, or constraints tying fields together at interfaces. The model's capacity to directly represent these fields stems from the ability of spline bases to express piecewise regularity and discontinuity.

Material discontinuities are handled solely by sampling the correct local $f(x)$1 (material tensor) at each collocation point and do not require modifying the network architecture or training strategy.

6. Training Protocol and Computational Efficiency

PIKAN is typically trained using L-BFGS (initial learning rate 1e-3) and may optionally use Adam warm-up. For 2D elasticity problems, collocation sets of $f(x)$2–$f(x)$3 interior and $f(x)$4 boundary points are standard. A single KAN model with $f(x)$5 parameters is used for the entire domain, offering substantial parameter and computational efficiency compared to multi-domain PINNs (e.g., CENN) which require multiple MLPs and numerous penalty hyperparameters. In practice, PIKAN achieves faster convergence and attains higher or equivalent accuracy relative to domain-decomposed PINNs, with the primary additional cost being the evaluation of B-splines during forward and backward passes.

7. Numerical Validation and Practical Findings

PIKAN was benchmarked on several multi-material electronic packaging elasticity problems:

• Cantilever beams with straight, wavy, and stepped interfaces: PIKAN (KAN [2,5,5,5,2], 1050 parameters) achieved sub-1% relative $f(x)$6 error vs. FEM, outperforming CENN by an order of magnitude in error.
• A square plate with a central hole and circular interface: relative displacement $f(x)$7 error $f(x)$8, maximum absolute error $f(x)$9.
• Direct-bonded-copper (DBC) and through-glass-via (TGV–Cu) structures: PIKAN provided accurate displacement and stress fields, with interface jumps accurately resolved and maximum errors in the $\bar t$0 range (or lower), achieved without explicit interface constraints.

Convergence of the energy loss was observed to be robust across all tested integration schemes, with triangular integration converging fastest. Optimal accuracy depends on the choice of B-spline grid and order, which must be co-tuned with KAN network depth.

In summary, PIKAN achieves high-accuracy, computationally efficient, mesh-free solution of multi-material elasticity problems by combining exact variational principle minimization, admissible field construction, and spline-based KAN architectures that inherently accommodate interface discontinuities (Gong et al., 23 Aug 2025). This approach is particularly potent in heterogeneous domains where the sharpness and location of material interfaces present major challenges for classical PINN and domain-decomposition-based PINN strategies.