Model Reduction and Neural Networks for Parametric PDEs

• URL: http://arxiv.org/abs/2005.03180v2
• Date: Thu, 17 Jun 2021 18:45:06 GMT
• Title: Model Reduction and Neural Networks for Parametric PDEs
• Authors: Kaushik Bhattacharya, Bamdad Hosseini, Nikola B. Kovachki, Andrew M. Stuart
• Abstract summary: We develop a framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology.
• Score: 9.405458160620533
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. We also include numerical experiments which demonstrate the effectiveness of the method, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare it with existing algorithms from the literature; our examples include the mapping from coefficient to solution in a divergence form elliptic partial differential equation (PDE) problem, and the solution operator for viscous Burgers' equation.

Related papers

• Shape-informed surrogate models based on signed distance function domain encoding [8.052704959617207]
  We propose a non-intrusive method to build surrogate models that approximate the solution of parameterized partial differential equations (PDEs) Our approach is based on the combination of two neural networks (NNs)
  arXiv  Detail & Related papers  (2024-09-19T01:47:04Z)
• Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models [50.90868087591973]
  We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation.
  arXiv  Detail & Related papers  (2024-08-20T19:06:02Z)
• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• Approximation of Solution Operators for High-dimensional PDEs [2.3076986663832044]
  We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations. Results are presented for several high-dimensional PDEs, including real-world applications to solving Hamilton-Jacobi-Bellman equations.
  arXiv  Detail & Related papers  (2024-01-18T21:45:09Z)
• Probabilistic partition of unity networks for high-dimensional regression problems [1.0227479910430863]
  We explore the partition of unity network (PPOU-Net) model in the context of high-dimensional regression problems. We propose a general framework focusing on adaptive dimensionality reduction. The PPOU-Nets consistently outperform the baseline fully-connected neural networks of comparable sizes in numerical experiments.
  arXiv  Detail & Related papers  (2022-10-06T06:01:36Z)
• An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations [68.8204255655161]
  Filtering equations play a central role in many real-life applications, including numerical weather prediction, finance and engineering. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method. We combine this method with a neural network representation to produce an approximation of the unnormalised conditional distribution of the signal process.
  arXiv  Detail & Related papers  (2022-01-10T11:01:36Z)
• A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations [0.2148535041822524]
  We develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields.
  arXiv  Detail & Related papers  (2021-03-10T17:01:42Z)
• Partition-based formulations for mixed-integer optimization of trained ReLU neural networks [66.88252321870085]
  This paper introduces a class of mixed-integer formulations for trained ReLU neural networks. At one extreme, one partition per input recovers the convex hull of a node, i.e., the tightest possible formulation for each node.
  arXiv  Detail & Related papers  (2021-02-08T17:27:34Z)
• Optimal oracle inequalities for solving projected fixed-point equations [53.31620399640334]
  We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation.
  arXiv  Detail & Related papers  (2020-12-09T20:19:32Z)
• The Random Feature Model for Input-Output Maps between Banach Spaces [6.282068591820945]
  The random feature model is a parametric approximation to kernel or regression methods. We propose a methodology for use of the random feature model as a data-driven surrogate for operators that map an input Banach space to an output Banach space.
  arXiv  Detail & Related papers  (2020-05-20T17:41:40Z)
• Neural Operator: Graph Kernel Network for Partial Differential Equations [57.90284928158383]
  This work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators) We formulate approximation of the infinite-dimensional mapping by composing nonlinear activation functions and a class of integral operators. Experiments confirm that the proposed graph kernel network does have the desired properties and show competitive performance compared to the state of the art solvers.
  arXiv  Detail & Related papers  (2020-03-07T01:56:20Z)

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.