DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems

TL;DR

The paper introduces DeepOKAN, a neural-operator framework that replaces DeepONet's MLP branches/trunks with Kolmogorov-Arnold networks (KANs) and incorporates Gaussian radial basis functions (RBFs) to learn input-output operators in mechanics. By formulating mappings $F: Q \to S$ with edge-activated KAN layers and an RBF-KAN implementation, the authors demonstrate improved accuracy and training stability over traditional DeepONet across 1D sinusoidal waves, 2D orthotropic elasticity, and transient Poisson problems, using RMSD/L2-error metrics. Key findings show DeepOKAN achieves lower training losses, tighter error distributions, and more accurate field predictions, including sharper high-frequency features and consistent performance across multiple seeds and network complexities. The work highlights the potential of KAN-based neural operators for efficient, robust surrogates in digital-twin and design workflows, with future directions including exploring alternate KAN bases, geometry handling, and broader physical applications.

Abstract

The modern digital engineering design often requires costly repeated simulations for different scenarios. The prediction capability of neural networks (NNs) makes them suitable surrogates for providing design insights. However, only a few NNs can efficiently handle complex engineering scenario predictions. We introduce a new version of the neural operators called DeepOKAN, which utilizes Kolmogorov Arnold networks (KANs) rather than the conventional neural network architectures. Our DeepOKAN uses Gaussian radial basis functions (RBFs) rather than the B-splines. RBFs offer good approximation properties and are typically computationally fast. The KAN architecture, combined with RBFs, allows DeepOKANs to represent better intricate relationships between input parameters and output fields, resulting in more accurate predictions across various mechanics problems. Specifically, we evaluate DeepOKAN's performance on several mechanics problems, including 1D sinusoidal waves, 2D orthotropic elasticity, and transient Poisson's problem, consistently achieving lower training losses and more accurate predictions compared to traditional DeepONets. This approach should pave the way for further improving the performance of neural operators.

Figures (18)

• Figure 1: Illustration of the activation functions flowing through the network.
• Figure 2: Visualization of RBF-KAN Layer: Each curve represents an individual basis function.
• Figure 3: Schematic for neural operators.
• Figure 4: Illustration of (a.) DeepONets and (b.) DeepOKANs. $\phi^t$ and $\phi^b$ represent the activation functions corresponding to the trunk and branch networks, respectively. $p$ and $v$ are hyperparameters defining the width of layer $l=1$ in the branch and trunk, respectively. Different layers can possess different widths..
• Figure 5: Results of the $1^{st}$ sinusoidal wave example (Wave-Case1): a. true and predicted values, b. evolution of the training loss.