Model Reduction And Neural Networks For Parametric PDEs

Abstract

We develop a general framework for data-driven approximation of input-output maps between infinitedimensional 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.

Calibration and Uncertainty Quantification of Convective Parameters in an Idealized GCM

“…However, other supervised machine learning techniques have potential to do so. For example, random feature maps and deep neural networks show promise in this regard (Bhattacharya et al, 2020;Nelsen & Stuart, 2020); incorporating these tools in the CES algorithm is a direction of current research.…”

A learning-based multiscale method and its application to inelastic impact problems

“…It will be useful if we can learn this as a part of the training, i.e., extend the map F : {{U (τ ) : τ ∈ (0, t)}, ξ 0 }} → σ (t) t ∈ (0, T ). This has been successfully demonstrated in simple problems like Darcy flow [7], and remains a work in progress.…”

“…Now, the implementation of the multiscale problem above requires the calculation of the map F, and therefore the unit cell problem at each macroscopic point x and at each instant t; this is extremely expensive. Our idea is to learn the macroscopic constitutive behavior using model reduction and deep neural networks following the approach of Bhattacharya et al [7] by utilizing data generated by solutions of the unit cell problem over various strain histories obtained from an appropriate probability distribution in the space of strain histories. To do so, we observe that the unit cell problem in fact specifies the map…”

“…where W 1 , ..., W M are the weight matrices, b 1 , ..., b M are the bias vectors, and ω is the scaled exponential linear unit (SELU) [30] with scaling constants χ = 1.67 and β = 1.05. We approximate the PCA specified maps by a standard SVD algorithm and the weight and bias parameters of the neural network with standard stochastic gradient based minimization techniques [7]. This learnt approximate map replaces the constitutive relation in a macroscopic integrator.…”

“…While typical approaches use a finite-dimensional subspace obtained by discretization to solve these problems, it is desirable for the learnt map to be independent of the particular discretization or resolution. Therefore we use the approach developed in Bhattacharya et al [7] that combines model reduction and neural networks for high-fidelity approximations of maps between function spaces.…”