Binary structured physics-informed neural networks for solving equations with rapidly changing solutions

• URL: http://arxiv.org/abs/2401.12806v2
• Date: Thu, 25 Jan 2024 12:53:39 GMT
• Title: Binary structured physics-informed neural networks for solving equations with rapidly changing solutions
• Authors: Yanzhi Liu and Ruifan Wu and Ying Jiang
• Abstract summary: 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.
• Score: 3.6415476576196055
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Physics-informed neural networks (PINNs), rooted in deep learning, have emerged as a promising approach for solving partial differential equations (PDEs). By embedding the physical information described by PDEs into feedforward neural networks, PINNs are trained as surrogate models to approximate solutions without the need for label data. Nevertheless, even though PINNs have shown remarkable performance, they can face difficulties, especially when dealing with equations featuring rapidly changing solutions. These difficulties encompass slow convergence, susceptibility to becoming trapped in local minima, and reduced solution accuracy. To address these issues, we propose a binary structured physics-informed neural network (BsPINN) framework, which employs binary structured neural network (BsNN) as the neural network component. By leveraging a binary structure that reduces inter-neuron connections compared to fully connected neural networks, BsPINNs excel in capturing the local features of solutions more effectively and efficiently. These features are particularly crucial for learning the rapidly changing in the nature of solutions. In a series of numerical experiments solving Burgers equation, Euler equation, Helmholtz equation, and high-dimension Poisson equation, BsPINNs exhibit superior convergence speed and heightened accuracy compared to PINNs. From these experiments, we discover that BsPINNs resolve the issues caused by increased hidden layers in PINNs resulting in over-smoothing, and prevent the decline in accuracy due to non-smoothness of PDEs solutions.

Related papers

• Ensemble learning for Physics Informed Neural Networks: a Gradient Boosting approach [10.250994619846416]
  We present a new training paradigm referred to as "gradient boosting" (GB) Instead of learning the solution of a given PDE using a single neural network directly, our algorithm employs a sequence of neural networks to achieve a superior outcome. This work also unlocks the door to employing ensemble learning techniques in PINNs.
  arXiv  Detail & Related papers  (2023-02-25T19:11:44Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• Characteristics-Informed Neural Networks for Forward and Inverse Hyperbolic Problems [0.0]
  We propose characteristic-informed neural networks (CINN) for solving forward and inverse problems involving hyperbolic PDEs. CINN encodes the characteristics of the PDE in a general-purpose deep neural network trained with the usual MSE data-fitting regression loss. Preliminary results indicate that CINN is able to improve on the accuracy of the baseline PINN, while being nearly twice as fast to train and avoiding non-physical solutions.
  arXiv  Detail & Related papers  (2022-12-28T18:38:53Z)
• RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network [4.425915683879297]
  We propose a novel framework based on graph neural networks (GNNs) and radial basis function finite difference (RBF-FD) RBF-FD is used to construct a high-precision difference format of the differential equations to guide model training. We illustrate the generalizability, accuracy, and efficiency of the proposed algorithms on different PDE parameters.
  arXiv  Detail & Related papers  (2022-12-06T10:08:02Z)
• 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)
• Improved Training of Physics-Informed Neural Networks with Model Ensembles [81.38804205212425]
  We propose to expand the solution interval gradually to make the PINN converge to the correct solution. All ensemble members converge to the same solution in the vicinity of observed data. We show experimentally that the proposed method can improve the accuracy of the found solution.
  arXiv  Detail & Related papers  (2022-04-11T14:05:34Z)
• 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)
• Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations [20.277873724720987]
  We propose a new, scalable approach for solving large problems relating to differential equations called Finite Basis PINNs (FBPINNs) FBPINNs are inspired by classical finite element methods, where the solution of the differential equation is expressed as the sum of a finite set of basis functions with compact support. In FBPINNs neural networks are used to learn these basis functions, which are defined over small, overlapping subdomain problems.
  arXiv  Detail & Related papers  (2021-07-16T13:03:47Z)
• 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.