Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks

• URL: http://arxiv.org/abs/2310.09528v1
• Date: Sat, 14 Oct 2023 08:13:43 GMT
• Title: Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks
• Authors: Woojin Cho, Kookjin Lee, Donsub Rim, Noseong Park
• Abstract summary: In this study, we suggest a path that potentially opens up a possibility for physics-informed neural networks (PINNs) to be considered as one such solver. PINNs have pioneered a proper integration of deep-learning and scientific computing, but they require repetitive time-consuming training of neural networks. We propose a lightweight low-rank PINNs containing only hundreds of model parameters and an associated hypernetwork-based meta-learning algorithm.
• Score: 24.14254861023394
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In various engineering and applied science applications, repetitive numerical simulations of partial differential equations (PDEs) for varying input parameters are often required (e.g., aircraft shape optimization over many design parameters) and solvers are required to perform rapid execution. In this study, we suggest a path that potentially opens up a possibility for physics-informed neural networks (PINNs), emerging deep-learning-based solvers, to be considered as one such solver. Although PINNs have pioneered a proper integration of deep-learning and scientific computing, they require repetitive time-consuming training of neural networks, which is not suitable for many-query scenarios. To address this issue, we propose a lightweight low-rank PINNs containing only hundreds of model parameters and an associated hypernetwork-based meta-learning algorithm, which allows efficient approximation of solutions of PDEs for varying ranges of PDE input parameters. Moreover, we show that the proposed method is effective in overcoming a challenging issue, known as "failure modes" of PINNs.

Related papers

• Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks [0.0]
  This work explores the potential usage of graph neural networks (GNNs) for the simulation of time-dependent PDEs. We propose a systematic strategy to build surrogate models based on a data-driven time-stepping scheme. We show that GNNs can provide a valid alternative to traditional surrogate models in terms of computational efficiency and generalization to new scenarios.
  arXiv  Detail & Related papers  (2023-08-03T08:14:28Z)
• A Stable and Scalable Method for Solving Initial Value PDEs with Neural Networks [52.5899851000193]
  We develop an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters. We show that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors.
  arXiv  Detail & Related papers  (2023-04-28T17:28:18Z)
• A physics-informed neural network framework for modeling obstacle-related equations [3.687313790402688]
  Physics-informed neural networks (PINNs) are an attractive tool for solving partial differential equations based on sparse and noisy data. Here we extend PINNs to solve obstacle-related PDEs which present a great computational challenge. The performance of the proposed PINNs is demonstrated in multiple scenarios for linear and nonlinear PDEs subject to regular and irregular obstacles.
  arXiv  Detail & Related papers  (2023-04-07T09:22:28Z)
• 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)
• 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)
• Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problems [0.0]
  In this work, we combine two classes of numerical methods: (i) physics-informed neural networks (PINNs) and (ii) adaptive spectral methods. The numerical methods that we develop take advantage of the ability of physics-informed neural networks to easily implement high-order numerical schemes to efficiently solve PDEs. We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs.
  arXiv  Detail & Related papers  (2022-02-06T05:25:22Z)
• 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)
• Optimal Transport Based Refinement of Physics-Informed Neural Networks [0.0]
  We propose a refinement strategy to the well-known Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs) based on the concept of Optimal Transport (OT) PINNs solvers have been found to suffer from a host of issues: spectral bias in fully-connected pathologies, unstable gradient, and difficulties with convergence and accuracy. We present a novel training strategy for solving the Fokker-Planck-Kolmogorov Equation (FPKE) using OT-based sampling to supplement the existing PINNs framework.
  arXiv  Detail & Related papers  (2021-05-26T02:51:20Z)
• 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)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)

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.