Source-linked AI summary
DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems
Diab W. Abueidda, Panos Pantidis, Mostafa E. Mobasher
TL;DR
The paper addresses the need for efficient surrogates for repeated, computationally intensive mechanics simulations. It introduces DeepOKAN, combining KAN-based DeepONet branch and trunk networks with Gaussian RBFs, and reports lower losses and more accurate predictions than DeepONet across several mechanics problems.
Problem
Repeated high-fidelity simulations are computationally intensive, while existing surrogate models may require retraining when loads, conditions, materials, or geometry change.
Method
DeepOKAN replaces DeepONet’s MLP branch and trunk networks with KANs, integrates RBF-based function approximation, and is evaluated under matched parameter-count and training settings.
Results
DeepOKAN consistently outperforms DeepONet in the reported mechanics examples, including lower training losses and more accurate predictions; for 2D orthotropic elasticity, mean L2-norm error is 0.0047 versus 0.02980.
Takeaways & Limitations
DeepOKAN provides a KAN- and RBF-based neural-operator approach for efficiently approximating solution fields across several mechanics problems.
Takeaways & Limitations
For orthotropic elasticity data generation, the study assumes a physically plausible range for Gxy using an equation that does not generally hold for orthotropic materials.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper motivates neural-operator surrogates for computationally intensive mechanics simulations and introduces DeepOKAN, which replaces DeepONet’s MLP branch and trunk networks with KANs and incorporates RBFs.
- Motivation: High-fidelity finite element simulations are computationally intensive because mechanics problems vary in loads, conditions, materials, geometries, physics, dimensionality, and time.These costs motivate surrogate models for repeated engineering evaluations.
- Motivation: Neural-network surrogates can rapidly infer physical solutions after training, supporting repeated predictions, optimization, sensitivity analysis, and uncertainty quantification.Their utility is tied to avoiding expensive numerical simulations during subsequent evaluations.
- Motivation: Existing surrogate models often require retraining or transfer learning when loads, boundary conditions, material properties, or geometry change.This limitation motivates operator-learning approaches for varying problem conditions.
- Related architectures: DeepONet maps parametric functions to solution spaces using branch and trunk subnetworks that were initially implemented as MLPs.The branch encodes input functions, while the trunk encodes domain geometry.
- Proposed approach: DeepOKAN replaces DeepONet’s MLP branch and trunk with KANs and integrates radial basis functions for flexible multivariate function approximation.The study applies this architecture to mechanics problems including sinusoidal waves, orthotropic elasticity, and transient Poisson’s problems.
2 Methods
The paper develops DeepOKAN by replacing DeepONet’s MLP branch and trunk networks with KANs, using Gaussian RBFs as learnable one-dimensional transformations. The resulting neural operator maps function inputs and domain coordinates to predicted PDE solution fields.
- 2.2 Kolmogorov-Arnold network: KAN layers place learnable one-dimensional activation functions on edges and sum their outputs to produce the next-layer activations.A KAN layer is represented as a matrix of functions connecting consecutive neurons.
- 2.2 Kolmogorov-Arnold network: Unlike MLPs, KANs combine linear transformations and nonlinearities within the function matrix Ψ.MLPs alternate separate linear transformations and fixed nonlinear activations.
- 2.3 RBF-KAN: RBF-KAN replaces the B-spline activation functions used in the original KAN implementation with Gaussian radial basis functions.Each Gaussian RBF depends on distance from a central point; its centers may be learnable or fixed.
- 2.3 RBF-KAN: RBF-KAN transforms inputs using basis functions centered on grid points, then combines the transformed features linearly with learnable weights.The resulting localized Gaussian-like responses are intended to capture complex nonlinear patterns.
- 2.4 Neural operators: Neural operators learn mappings from input functions q to PDE solution functions s(q), using branch outputs for inputs and trunk outputs for domain positions.DeepONet and DeepOKAN fuse branch and trunk representations to predict the solution operator at domain points.
- 2.5 DeepOKAN vs. DeepONet: DeepOKAN replaces the MLP branch and trunk networks of the DeepONet baseline with KANs, specifically Gaussian RBF KANs.The framework could also be extended to other KAN variants, including multi-quadratic RBF and B-spline KANs.
3 Numerical examples
The numerical examples compare RBF-KAN-based DeepOKANs with MLP-based DeepONets across sinusoidal waves, orthotropic elasticity, and transient Poisson problems. DeepOKANs generally show smoother or lower training losses and smaller prediction errors under matched evaluation settings.
- Evaluation setup: The evaluation matches trunk and branch depth, output width, and learnable parameter count between DeepOKAN and DeepONet.Hidden-layer width is varied to obtain comparable parameter counts while preserving other expressivity-related aspects.
- 1D sinusoidal waves: RBF-KAN approximates the first sinusoidal wave more accurately than MLP, particularly in high-frequency regions, with a greater loss reduction.The comparison uses two hidden layers and closely matched parameter counts.
- 1D sinusoidal waves: For the more oscillatory wave, RBF-KAN matches the analytical solution better than MLP, which fails to capture high-frequency regimes despite matched weights and training epochs.The comparison includes Adam training at learning rates 10^-2 and 10^-3.
- 1D sinusoidal waves: DeepOKAN converges faster and more stably than DeepONet across both learning-rate settings, reaching lower training losses with fewer spikes.DeepONet shows higher spikes at 10^-2, while both models become more stable at 10^-3.
- 2D orthotropic elasticity: In orthotropic elasticity, DeepOKAN attains smaller L2-norm errors than DeepONet across seeds and network complexities, with more accurate absolute-error maps.The comparison assumes zero body force and negligible inertia; the dataset contains 5000 samples split 80%/20%.
- 2D orthotropic elasticity: For the high-complexity orthotropic-elasticity models, mean L2-norm error is 0.0047 for DeepOKAN versus 0.02980 for DeepONet.DeepONet also exhibits greater variability and more outliers in the testing analysis.
4 Conclusions, limitations, and future work
DeepOKAN replaces DeepONet’s MLP branch and trunk with KANs using Gaussian RBFs as activation-function bases, and the proposed operator outperforms the classical data-driven DeepONet framework.
- DeepOKAN uses KANs in both branch and trunk components instead of the MLP architecture used by DeepONet.
- Gaussian RBFs serve as the basis functions for the learnable KAN activations in both DeepOKAN components.
- DeepOKAN is designed to approximate PDE solution mappings between function spaces for complex, high-dimensional physics and engineering problems.
- The proposed DeepOKAN outperforms the classical data-driven DeepONet framework.