The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach

• URL: http://arxiv.org/abs/2302.08309v2
• Date: Sun, 28 Jul 2024 10:52:47 GMT
• Title: The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approach
• Authors: Yongcun Song, Xiaoming Yuan, Hangrui Yue,
• Abstract summary: We study the combination of the alternating direction method of multipliers (ADMM) with physics-informed neural networks (PINNs) The resulting ADMM-PINNs algorithmic framework substantially enlarges the applicable range of PINNs to nonsmooth cases of PDE-constrained optimization problems. We validate the efficiency of the ADMM-PINNs algorithmic framework by different prototype applications.
• Score: 1.9030954416586594
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the combination of the alternating direction method of multipliers (ADMM) with physics-informed neural networks (PINNs) for a general class of nonsmooth partial differential equation (PDE)-constrained optimization problems, where additional regularization can be employed for constraints on the control or design variables. The resulting ADMM-PINNs algorithmic framework substantially enlarges the applicable range of PINNs to nonsmooth cases of PDE-constrained optimization problems. The application of the ADMM makes it possible to untie the PDE constraints and the nonsmooth regularization terms for iterations. Accordingly, at each iteration, one of the resulting subproblems is a smooth PDE-constrained optimization which can be efficiently solved by PINNs, and the other is a simple nonsmooth optimization problem which usually has a closed-form solution or can be efficiently solved by various standard optimization algorithms or pre-trained neural networks. The ADMM-PINNs algorithmic framework does not require to solve PDEs repeatedly, and it is mesh-free, easy to implement, and scalable to different PDE settings. We validate the efficiency of the ADMM-PINNs algorithmic framework by different prototype applications, including inverse potential problems, source identification in elliptic equations, control constrained optimal control of the Burgers equation, and sparse optimal control of parabolic equations.

Related papers

• A Simulation-Free Deep Learning Approach to Stochastic Optimal Control [12.699529713351287]
  We propose a simulation-free algorithm for the solution of generic problems in optimal control (SOC) Unlike existing methods, our approach does not require the solution of an adjoint problem.
  arXiv  Detail & Related papers  (2024-10-07T16:16:53Z)
• RoPINN: Region Optimized Physics-Informed Neural Networks [66.38369833561039]
  Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) This paper proposes and theoretically studies a new training paradigm as region optimization. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm.
  arXiv  Detail & Related papers  (2024-05-23T09:45:57Z)
• Analyzing and Enhancing the Backward-Pass Convergence of Unrolled Optimization [50.38518771642365]
  The integration of constrained optimization models as components in deep networks has led to promising advances on many specialized learning tasks. A central challenge in this setting is backpropagation through the solution of an optimization problem, which often lacks a closed form. This paper provides theoretical insights into the backward pass of unrolled optimization, showing that it is equivalent to the solution of a linear system by a particular iterative method. A system called Folded Optimization is proposed to construct more efficient backpropagation rules from unrolled solver implementations.
  arXiv  Detail & Related papers  (2023-12-28T23:15:18Z)
• Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators [6.296120102486062]
  We propose a novel framework for solving large-scale partial differential equations (PDEs) with high-dimensional random parameters. We refer to such neural operators as multi-input reduced basis derivative informed neural operators (MR-DINOs) We show that MR-DINOs offer $103$--$107 times$ reductions in execution time, and are able to produce OUU solutions of comparable accuracies to those from standard PDE based solutions.
  arXiv  Detail & Related papers  (2023-05-31T17:26:20Z)
• Bi-level Physics-Informed Neural Networks for PDE Constrained Optimization using Broyden's Hypergradients [29.487375792661005]
  We present a novel bi-level optimization framework to solve PDE constrained optimization problems. For the inner loop optimization, we adopt PINNs to solve the PDE constraints only. For the outer loop, we design a novel method by using Broyden'simat method based on the Implicit Function Theorem.
  arXiv  Detail & Related papers  (2022-09-15T06:21:24Z)
• Neural Stochastic Dual Dynamic Programming [99.80617899593526]
  We introduce a trainable neural model that learns to map problem instances to a piece-wise linear value function. $nu$-SDDP can significantly reduce problem solving cost without sacrificing solution quality.
  arXiv  Detail & Related papers  (2021-12-01T22:55:23Z)
• Faster Algorithm and Sharper Analysis for Constrained Markov Decision Process [56.55075925645864]
  The problem of constrained decision process (CMDP) is investigated, where an agent aims to maximize the expected accumulated discounted reward subject to multiple constraints. A new utilities-dual convex approach is proposed with novel integration of three ingredients: regularized policy, dual regularizer, and Nesterov's gradient descent dual. This is the first demonstration that nonconcave CMDP problems can attain the lower bound of $mathcal O (1/epsilon)$ for all complexity optimization subject to convex constraints.
  arXiv  Detail & Related papers  (2021-10-20T02:57:21Z)
• Physics and Equality Constrained Artificial Neural Networks: Application to Partial Differential Equations [1.370633147306388]
  Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE) Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach. We propose a versatile framework that can tackle both inverse and forward problems.
  arXiv  Detail & Related papers  (2021-09-30T05:55:35Z)
• Speeding up Computational Morphogenesis with Online Neural Synthetic Gradients [51.42959998304931]
  A wide range of modern science and engineering applications are formulated as optimization problems with a system of partial differential equations (PDEs) as constraints. These PDE-constrained optimization problems are typically solved in a standard discretize-then-optimize approach. We propose a general framework to speed up PDE-constrained optimization using online neural synthetic gradients (ONSG) with a novel two-scale optimization scheme.
  arXiv  Detail & Related papers  (2021-04-25T22:43:51Z)
• 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.