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

现代数字工程设计通常需要昂贵的重复模拟来应对不同的情况。神经网络（NN）的预测能力使它们成为提供设计洞察的合适替代品。然而，只有少数NN可以有效地处理复杂的工程场景预测。我们介绍了神经算子的一个新版本，称为DeepOKAN，它使用Kolmogorov Arnold网络（KAN）而不是传统的神经网络架构。我们的DeepOKAN使用高斯径向基函数（RBFs）而不是B样条。RBF具有良好的逼近性质，通常计算速度较快。KAN架构结合RBF，使DeepOKAN能够更好地表示输入参数和输出场之间的错综复杂关系，从而在各种力学问题中实现更准确的预测。具体来说，我们评估了DeepOKAN在几个力学问题上的性能，包括1D正弦波、2D正交弹性和瞬态泊松问题，相对于传统的DeepONets，它始终实现更低的训练损失和更准确的预测。这种方法应该为进一步提高神经算子的性能铺平道路。

DeepOKAN: A Deep Learning-Based Operator for Solving Complex Mechanics Problems

The paper proposes a new version of neural operators called DeepOKAN which utilizes Kolmogorov Arnold networks (KANs) and Gaussian radial basis functions (RBFs) to provide accurate predictions for complex engineering scenarios.

DeepOKAN consistently achieves lower training losses and more accurate predictions compared to traditional DeepONets across several mechanics problems, including 1D sinusoidal waves, 2D orthotropic elasticity, and transient Poisson's problem. The use of KANs and RBFs allows for better representation of intricate relationships between input parameters and output fields. The approach has the potential to improve the performance of neural operators in the future.

Recent related studies include 'DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators' by Lu et al. and 'Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations' by Raissi et al.