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

ML LFDDR NN AI4S PINNs
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。
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.
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