Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks

• URL: http://arxiv.org/abs/2406.11045v2
• Date: Sun, 4 Aug 2024 05:12:25 GMT
• Title: Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks
• Authors: Yizheng Wang, Jia Sun, Jinshuai Bai, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, Yinghua Liu,
• Abstract summary: AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs) The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to enhance the previously computational convergence-based PINNs. We propose different PDE forms based on KAN instead of, termed Kolmogorov-Arnold Neural Network (KINN) for solving forward and inverse problems.
• Score: 1.4053129774629072
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN) for solving forward and inverse problems. We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP regarding accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

Related papers

• SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networks [0.9999629695552196]
  Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) We introduce Separable Physics-Informed Kolmogorov-Arnold Networks (SPIKANs) This novel architecture applies the principle of separation of variables to PIKANs, decomposing the problem such that each dimension is handled by an individual KAN.
  arXiv  Detail & Related papers  (2024-11-09T21:10:23Z)
• Unisolver: PDE-Conditional Transformers Are Universal PDE Solvers [55.0876373185983]
  We present the Universal PDE solver (Unisolver) capable of solving a wide scope of PDEs. Our key finding is that a PDE solution is fundamentally under the control of a series of PDE components. Unisolver achieves consistent state-of-the-art results on three challenging large-scale benchmarks.
  arXiv  Detail & Related papers  (2024-05-27T15:34:35Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• Reduced-order modeling for parameterized PDEs via implicit neural representations [4.135710717238787]
  We present a new data-driven reduced-order modeling approach to efficiently solve parametrized partial differential equations (PDEs) The proposed framework encodes PDE and utilizes a parametrized neural ODE (PNODE) to learn latent dynamics characterized by multiple PDE parameters. We evaluate the proposed method at a large Reynolds number and obtain up to speedup of O(103) and 1% relative error to the ground truth values.
  arXiv  Detail & Related papers  (2023-11-28T01:35:06Z)
• Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks [17.69666422395703]
  Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere.
  arXiv  Detail & Related papers  (2023-08-18T14:58:23Z)
• Physics-Aware Neural Networks for Boundary Layer Linear Problems [0.0]
  Physics-Informed Neural Networks (PINNs) approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost of a Neural Network. This paper explores PINNs for linear PDEs whose solutions may present one or more boundary layers.
  arXiv  Detail & Related papers  (2022-07-15T21:15:06Z)
• Revisiting PINNs: Generative Adversarial Physics-informed Neural Networks and Point-weighting Method [70.19159220248805]
  Physics-informed neural networks (PINNs) provide a deep learning framework for numerically solving partial differential equations (PDEs) We propose the generative adversarial neural network (GA-PINN), which integrates the generative adversarial (GA) mechanism with the structure of PINNs. Inspired from the weighting strategy of the Adaboost method, we then introduce a point-weighting (PW) method to improve the training efficiency of PINNs.
  arXiv  Detail & Related papers  (2022-05-18T06:50:44Z)
• Learning Physics-Informed Neural Networks without Stacked Back-propagation [82.26566759276105]
  We develop a novel approach that can significantly accelerate the training of Physics-Informed Neural Networks. In particular, we parameterize the PDE solution by the Gaussian smoothed model and show that, derived from Stein's Identity, the second-order derivatives can be efficiently calculated without back-propagation. Experimental results show that our proposed method can achieve competitive error compared to standard PINN training but is two orders of magnitude faster.
  arXiv  Detail & Related papers  (2022-02-18T18:07:54Z)
• Lie Point Symmetry Data Augmentation for Neural PDE Solvers [69.72427135610106]
  We present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity. In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations. We show how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.
  arXiv  Detail & Related papers  (2022-02-15T18:43:17Z)
• dNNsolve: an efficient NN-based PDE solver [62.997667081978825]
  We introduce dNNsolve, that makes use of dual Neural Networks to solve ODEs/PDEs. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions.
  arXiv  Detail & Related papers  (2021-03-15T19:14:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.