🔢 mathematics

Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging

This paper proposes a Physics-Informed Kolmogorov-Arnold Network framework that leverages trainable B-spline activation functions to accurately and efficiently solve multi-material elasticity problems in electronic packaging using a single network, thereby eliminating the need for complex domain decomposition and interface constraints required by traditional methods.

Original authors: Yanpeng Gong, Yida He, Yue Mei, Xiaoying Zhuang, Fei Qin, Timon Rabczuk

Published 2026-02-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yanpeng Gong, Yida He, Yue Mei, Xiaoying Zhuang, Fei Qin, Timon Rabczuk

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Fixing the "Jigsaw Puzzle" Problem

Imagine you are trying to predict how a complex electronic package (like the chip inside your phone) will bend or stretch when you push on it. These packages are made of different materials glued together—like layers of copper, ceramic, and glass.

In the past, computer scientists used a method called Finite Element Method (FEM) to solve this. Think of FEM like trying to solve a puzzle by cutting the picture into thousands of tiny, rigid squares (a mesh). If the materials change abruptly (like going from soft copper to hard ceramic), the puzzle pieces have to be cut very precisely along the boundary, or the picture gets distorted. It's accurate, but it's slow and requires a lot of manual work to cut the pieces right.

Then, a new method called Physics-Informed Neural Networks (PINNs) arrived. Instead of cutting the picture into squares, PINNs use a "smart guesser" (a neural network) to learn the shape of the bend directly. However, standard smart guessers (called MLPs) are like smooth painters; they struggle to draw sharp lines or sudden changes. When two materials meet, the stress often jumps suddenly. A smooth painter blurs that line, leading to errors. To fix this, previous methods had to cut the domain into separate zones and force the two zones to agree at the border, which is like hiring two different painters and constantly yelling at them to match their brushstrokes at the seam.

The New Solution: The "Shape-Shifting" Network

This paper introduces a new tool called PIKAN (Physics-Informed Kolmogorov-Arnold Network).

1. The New Artist: KANs
The authors replaced the standard "smooth painter" neural network with a Kolmogorov-Arnold Network (KAN).

The Analogy: Imagine a standard neural network is a rigid ruler; it tries to draw a line by connecting dots with straight segments or smooth curves. A KAN is like a flexible, shape-shifting ruler made of small, adjustable segments (called B-splines).
Why it matters: Because these segments can change shape locally, the KAN can naturally draw a sharp corner or a sudden jump in the line without needing to be told to stop and start a new drawing. It handles the "rough edges" where different materials meet automatically, without needing to cut the problem into separate pieces.

2. The Strategy: The Deep Energy Method
The paper uses a specific way of teaching this network called the Deep Energy Method (DEM).

The Analogy: Imagine you are trying to find the lowest point in a foggy valley (the solution). Instead of trying to solve a complex equation for every single point, the network tries to minimize the total "energy" of the system.
The Benefit: The authors found that by building the network to automatically respect the rules of the edges (like holding the left side of the beam still), they didn't need to add extra "penalty" rules to the training. It's like building a slide that naturally ends at the ground, rather than building a slide in the air and hoping the kids don't fall off.

What They Tested

The researchers tested this new "shape-shifting" network on several scenarios:

Bending Beams: They simulated beams made of two or three different materials with straight, wavy, and stepped interfaces.
Holes in Plates: They tested a plate with a hole in the middle, made of two different materials.
Real-World Electronics: They applied it to a DBC substrate (a ceramic sandwiched between copper layers used in power electronics) and a TGV structure (glass with copper pillars used in advanced packaging).

The Results: Smoother, Faster, and More Accurate

When they compared PIKAN to the old methods (like the "two-painter" approach):

Accuracy: PIKAN predicted how the materials would bend and where the stress would be highest with much higher accuracy. It didn't get confused by the sharp boundaries between materials.
Simplicity: They only needed one single network to solve the whole problem, regardless of how many materials were inside. The old methods needed multiple networks and complex rules to make them talk to each other.
Efficiency: The new method required fewer "parameters" (the internal settings of the network) to achieve the same or better results than the old methods.

The Bottom Line

This paper shows that by using a new type of neural network (KAN) that is naturally good at handling sharp changes, combined with an energy-based learning strategy, we can simulate complex, multi-material electronic packages much more easily and accurately. It removes the need for complicated "domain decomposition" (cutting the problem into pieces) and lets the computer figure out the physics of the material boundaries on its own.

Note on Limitations: The paper explicitly states that while this works great for 2D examples (flat slices), extending it to full 3D objects is a future challenge that requires more research. They also note that the theoretical math behind why KANs work so well is still being fully explored.

Technical Summary: Physics-Informed Kolmogorov-Arnold Networks for Multi-Material Elasticity Problems in Electronic Packaging

Problem Statement
Electronic packaging structures consist of heterogeneous materials with distinct mechanical properties, creating complex multi-material systems characterized by sharp interfaces and material property discontinuities. Traditional numerical methods, such as the Finite Element Method (FEM), face challenges regarding mesh distortion, local field resolution, and computational efficiency when modeling these complex geometries. While Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative, standard implementations using Multi-Layer Perceptrons (MLPs) struggle with multi-material problems. Conventional PINN approaches typically require domain decomposition (partitioning the domain into subdomains for each material) and the introduction of interface continuity constraints via penalty terms. This necessitates extensive hyperparameter tuning and often leads to convergence issues, particularly for complex interface geometries. Furthermore, MLPs can suffer from spectral bias, limiting their ability to capture high-frequency features and sharp transitions inherent in multi-material solutions.

Methodology
This paper proposes a Physics-Informed Kolmogorov-Arnold Network (PIKAN) framework integrated with the Deep Energy Method (DEM) to solve multi-material elasticity problems without domain decomposition.

Deep Energy Method (DEM): Instead of solving the strong form of the governing partial differential equations (PDEs), the method minimizes the total potential energy functional. The loss function is constructed as the difference between internal strain energy and external potential energy. This variational approach reduces the order of derivatives required and typically offers higher accuracy and efficiency for elasticity problems.
Admissible Displacement Fields: To strictly satisfy essential boundary conditions without penalty terms, the displacement field is constructed as $u_{pred}(x) = P(x) + D(x) \odot F(x; \theta)$ , where $P(x)$ satisfies boundary values, $D(x)$ is a distance function to the boundary, and $F(x; \theta)$ is the neural network output. This ensures boundary conditions are met by construction.
Kolmogorov-Arnold Networks (KANs): The core innovation replaces traditional MLPs with KANs. Unlike MLPs, which use fixed activation functions at nodes and learnable weights on edges, KANs place learnable univariate activation functions (parameterized as B-splines) on the edges.
- Handling Discontinuities: The B-spline basis functions possess local support and piecewise characteristics. This allows a single KAN to naturally approximate discontinuities in stress and strain gradients across material interfaces without requiring separate networks for different materials or explicit interface continuity constraints.
- Architecture: The network employs trainable B-spline coefficients, scaling factors, and optional linear weights. Input coordinates are normalized to fit the B-spline definition domain, and intermediate layer outputs are constrained using $\tanh$ functions to maintain stability.
Numerical Integration: The study evaluates various numerical integration schemes (Triangular, Simpson, Trapezoidal, Monte Carlo, and Delaunay) to approximate the energy integrals in the loss function. The choice of integration scheme is shown to significantly impact solution accuracy, particularly in regions with high stress gradients.

Key Contributions

PIKAN Framework for Multi-Materials: The development of a PIKAN methodology that integrates KANs into the Deep Energy Method, leveraging the inherent piecewise nature of B-splines to accommodate material property discontinuities naturally.
Elimination of Domain Decomposition: The demonstration that a single KAN can approximate displacement fields across an entire multi-material domain, removing the need for subdomain partitioning and the associated interface continuity penalty terms and hyperparameter tuning.
Comprehensive Validation: Extensive numerical experiments on cantilever beams with straight, wavy, and stepped interfaces, a heterogeneous plate with a central hole, and electronic packaging structures (DBC substrates and TGV-Cu structures).
Systematic Analysis: A detailed investigation into the sensitivity of numerical integration schemes and hyperparameters, providing implementation details and source code for reproducibility.

Results

Accuracy and Robustness: PIKAN demonstrated excellent agreement with FEM reference solutions across all tested multi-material configurations. It successfully captured complex displacement fields and stress concentrations near interfaces, even with extreme contrasts in elastic moduli (e.g., spanning two orders of magnitude).
Comparison with Domain Decomposition: When compared to the Conservative Energy Neural Network (CENN), which uses domain decomposition with separate MLPs, PIKAN showed superior accuracy. CENN exhibited noticeable displacement discontinuities and stress oscillations at interfaces, particularly in complex geometries (wavy and stepped interfaces), due to the challenges of enforcing continuity via penalty terms.
Integration Scheme Sensitivity: Triangular integration generally provided the fastest convergence and highest accuracy. Delaunay integration showed weaker convergence and higher errors, particularly for curved interfaces, highlighting the critical role of the integration scheme in variational PINNs.
Parameter Efficiency: PIKAN achieved high accuracy with fewer trainable parameters compared to the MLP-based CENN approach. For instance, in the cantilever beam examples, PIKAN used approximately 1,050 parameters versus over 3,500 for CENN.
Electronic Packaging Applications: The method was successfully applied to analyze warping in DBC substrates and stress distributions in Through Glass Via (TGV) structures, including cases with interface roughness and dual-via configurations, confirming its practical utility.

Significance and Claims
The paper claims that PIKAN offers a more concise and robust solution paradigm for multi-material solid mechanics problems compared to traditional domain decomposition methods. By utilizing the piecewise approximation capabilities of KANs within an energy-based framework, the method simplifies the computational framework by eliminating the need for complex interface constraint handling and hyperparameter tuning. The authors posit that this approach is particularly well-suited for electronic packaging analysis, where multi-material interfaces are ubiquitous.

The study acknowledges that while the theoretical convergence of KANs is still an active research frontier, the empirical performance in these elasticity problems is promising. The authors note that the current work is limited to 2D examples and that extending the method to 3D structures presents challenges related to computational scalability and the theoretical limitations of standard KANs in representing 3D physical laws (E(3) transformations). Future work is suggested to explore hybrid MLP-KAN architectures and rigorous mathematical convergence proofs. The paper concludes that PIKAN represents a significant step forward in applying physics-informed deep learning to heterogeneous material problems, offering a pathway for more efficient and accurate reliability analysis in electronic packaging.

The Big Picture: Fixing the "Jigsaw Puzzle" Problem

The New Solution: The "Shape-Shifting" Network

What They Tested

The Results: Smoother, Faster, and More Accurate

The Bottom Line

More like this