Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems

• URL: http://arxiv.org/abs/2307.00296v2
• Date: Tue, 25 Jul 2023 17:15:23 GMT
• Title: Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems
• Authors: Yongcun Song, Xiaoming Yuan, Hangrui Yue
• Abstract summary: We focus on the application of a primal-dual method, with which different types of variables can be treated individually. For the accelerated primal-dual method with larger step sizes, its convergence can be proved rigorously while it numerically accelerates the original primal-dual method. For the operator learning acceleration, we construct deep neural network surrogate models for the involved PDEs.
• Score: 3.1006429989273063
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider a general class of nonsmooth optimal control problems with partial differential equation (PDE) constraints, which are very challenging due to its nonsmooth objective functionals and the resulting high-dimensional and ill-conditioned systems after discretization. We focus on the application of a primal-dual method, with which different types of variables can be treated individually and thus its main computation at each iteration only requires solving two PDEs. Our target is to accelerate the primal-dual method with either larger step sizes or operator learning techniques. For the accelerated primal-dual method with larger step sizes, its convergence can be still proved rigorously while it numerically accelerates the original primal-dual method in a simple and universal way. For the operator learning acceleration, we construct deep neural network surrogate models for the involved PDEs. Once a neural operator is learned, solving a PDE requires only a forward pass of the neural network, and the computational cost is thus substantially reduced. The accelerated primal-dual method with operator learning is mesh-free, numerically efficient, and scalable to different types of PDEs. The acceleration effectiveness of these two techniques is promisingly validated by some preliminary numerical results.

Related papers

• DOF: Accelerating High-order Differential Operators with Forward Propagation [40.71528485918067]
  We propose an efficient framework, Differential Operator with Forward-propagation (DOF), for calculating general second-order differential operators without losing any precision. We demonstrate two times improvement in efficiency and reduced memory consumption on any architectures. Empirical results illustrate that our method surpasses traditional automatic differentiation (AutoDiff) techniques, achieving 2x improvement on the structure and nearly 20x improvement on the sparsity.
  arXiv  Detail & Related papers  (2024-02-15T05:59:21Z)
• TSONN: Time-stepping-oriented neural network for solving partial differential equations [1.9061608251056779]
  This work integrates time-stepping method with deep learning to solve PDE problems. The convergence of model training is significantly improved by following the trajectory of the pseudo time-stepping process. Our results show that the proposed method achieves stable training and correct results in many problems that standard PINNs fail to solve.
  arXiv  Detail & Related papers  (2023-10-25T09:19:40Z)
• Multi-Grade Deep Learning for Partial Differential Equations with Applications to the Burgers Equation [3.5994228506864405]
  We develop in this paper a multi-grade deep learning method for solving nonlinear partial differential equations (PDEs) Deep neural networks (DNNs) have received super performance in solving PDEs. implementation in this paper focuses only on the 1D, 2D, and 3D Burgers equations.
  arXiv  Detail & Related papers  (2023-09-14T03:09:58Z)
• An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions [1.2981626828414923]
  We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. We present ample numerical simulations for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to demonstrate their performance.
  arXiv  Detail & Related papers  (2023-09-13T15:59:02Z)
• 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)
• iPINNs: Incremental learning for Physics-informed neural networks [66.4795381419701]
  Physics-informed neural networks (PINNs) have recently become a powerful tool for solving partial differential equations (PDEs) We propose incremental PINNs that can learn multiple tasks sequentially without additional parameters for new tasks and improve performance for every equation in the sequence. Our approach learns multiple PDEs starting from the simplest one by creating its own subnetwork for each PDE and allowing each subnetwork to overlap with previously learnedworks.
  arXiv  Detail & Related papers  (2023-04-10T20:19:20Z)
• Entropic Neural Optimal Transport via Diffusion Processes [105.34822201378763]
  We propose a novel neural algorithm for the fundamental problem of computing the entropic optimal transport (EOT) plan between continuous probability distributions. Our algorithm is based on the saddle point reformulation of the dynamic version of EOT which is known as the Schr"odinger Bridge problem. In contrast to the prior methods for large-scale EOT, our algorithm is end-to-end and consists of a single learning step.
  arXiv  Detail & Related papers  (2022-11-02T14:35:13Z)
• 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)
• 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.