KP-PINNs: Kernel Packet Accelerated Physics Informed Neural Networks

• URL: http://arxiv.org/abs/2506.08563v2
• Date: Wed, 11 Jun 2025 08:00:24 GMT
• Title: KP-PINNs: Kernel Packet Accelerated Physics Informed Neural Networks
• Authors: Siyuan Yang, Cheng Song, Zhilu Lai, Wenjia Wang,
• Abstract summary: We propose a new PINNs framework named Kernel Packet accelerated PINNs (KP-PINNs)<n>KP-PINNs gives a new expression of the loss function using the reproducing kernel Hilbert space (RKHS) norm and uses the Kernel Packet method to accelerate the computation.<n> Numerical experiments illustrate that KP-PINNs can solve differential equations effectively and efficiently.
• Score: 12.73776469872022
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Differential equations are involved in modeling many engineering problems. Many efforts have been devoted to solving differential equations. Due to the flexibility of neural networks, Physics Informed Neural Networks (PINNs) have recently been proposed to solve complex differential equations and have demonstrated superior performance in many applications. While the L2 loss function is usually a default choice in PINNs, it has been shown that the corresponding numerical solution is incorrect and unstable for some complex equations. In this work, we propose a new PINNs framework named Kernel Packet accelerated PINNs (KP-PINNs), which gives a new expression of the loss function using the reproducing kernel Hilbert space (RKHS) norm and uses the Kernel Packet (KP) method to accelerate the computation. Theoretical results show that KP-PINNs can be stable across various differential equations. Numerical experiments illustrate that KP-PINNs can solve differential equations effectively and efficiently. This framework provides a promising direction for improving the stability and accuracy of PINNs-based solvers in scientific computing.

Related papers

• Densely Multiplied Physics Informed Neural Networks [1.8554335256160261]
  physics-informed neural networks (PINNs) have shown great potential in dealing with nonlinear partial differential equations (PDEs) This paper improves the neural network architecture to improve the performance of PINN. We propose a densely multiply PINN (DM-PINN) architecture, which multiplies the output of a hidden layer with the outputs of all the behind hidden layers.
  arXiv  Detail & Related papers  (2024-02-06T20:45:31Z)
• Binary structured physics-informed neural networks for solving equations with rapidly changing solutions [3.6415476576196055]
  Physics-informed neural networks (PINNs) have emerged as a promising approach for solving partial differential equations (PDEs) We propose a binary structured physics-informed neural network (BsPINN) framework, which employs binary structured neural network (BsNN) as the neural network component. BsPINNs exhibit superior convergence speed and heightened accuracy compared to PINNs.
  arXiv  Detail & Related papers  (2024-01-23T14:37:51Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• $Δ$-PINNs: physics-informed neural networks on complex geometries [3.238149275474964]
  Physics-informed neural networks (PINNs) have demonstrated promise in solving forward and inverse problems involving partial differential equations.<n>To date, there is no clear way to inform PINNs about the topology of the domain where the problem is being solved.<n>We propose a novel positional encoding mechanism for PINNs based on the eigenfunctions of the Laplace-Beltrami operator.
  arXiv  Detail & Related papers  (2022-09-08T18:03:19Z)
• Reduced-PINN: An Integration-Based Physics-Informed Neural Networks for Stiff ODEs [0.0]
  Physics-informed neural networks (PINNs) have recently received much attention due to their capabilities in solving both forward and inverse problems. We propose a new PINN architecture, called Reduced-PINN, that utilizes a reduced-order integration method to enable the PINN to solve stiff chemical kinetics.
  arXiv  Detail & Related papers  (2022-08-23T09:20:42Z)
• Enforcing Continuous Physical Symmetries in Deep Learning Network for Solving Partial Differential Equations [3.6317085868198467]
  We introduce a new method, symmetry-enhanced physics informed neural network (SPINN) where the invariant surface conditions induced by the Lie symmetries of PDEs are embedded into the loss function of PINN. We show that SPINN performs better than PINN with fewer training points and simpler architecture of neural network.
  arXiv  Detail & Related papers  (2022-06-19T00:44:22Z)
• Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecture [77.59766598165551]
  Physics-informed neural networks (PINNs) are revolutionizing science and engineering practice by bringing together the power of deep learning to bear on scientific computation. Here, we propose Auto-PINN, which employs Neural Architecture Search (NAS) techniques to PINN design. A comprehensive set of pre-experiments using standard PDE benchmarks allows us to probe the structure-performance relationship in PINNs.
  arXiv  Detail & Related papers  (2022-05-27T03:24:31Z)
• 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)
• Characterizing possible failure modes in physics-informed neural networks [55.83255669840384]
  Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena even for simple PDEs. We show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize.
  arXiv  Detail & Related papers  (2021-09-02T16:06:45Z)
• 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.