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Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach

Ignacio Brevis, Ignacio Muga, David Pardo, Oscar Rodriguez, Kristoffer G. van der Zee

TL;DR

This work develops a neural-network–driven weighted Minimal-Residual Galerkin framework for parametric PDEs that targets accurate quantities of interest on coarse discretizations. By learning a λ-dependent weight ω(λ) to define a weighted inner product, the method tailors the MinRes discretization toward the QoI, while preserving the Galerkin structure. An affine-decomposition strategy enables preassembly of Gram, stiffness, and load matrices, enabling efficient offline training and fast online QoI evaluation. An adaptive training set further improves convergence and mitigates overfitting, with numerical results across 1D and 2D problems showing high QoI accuracy on coarse meshes and substantial computational efficiency gains.

Abstract

The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a parametric PDE. The central component in our approach is an efficient neural-network-weighted Minimal-Residual formulation, which, after training, provides Galerkin-based approximations in standard discrete spaces that have accurate quantities of interest, regardless of the coarseness of the discrete space.

Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach

TL;DR

This work develops a neural-network–driven weighted Minimal-Residual Galerkin framework for parametric PDEs that targets accurate quantities of interest on coarse discretizations. By learning a λ-dependent weight ω(λ) to define a weighted inner product, the method tailors the MinRes discretization toward the QoI, while preserving the Galerkin structure. An affine-decomposition strategy enables preassembly of Gram, stiffness, and load matrices, enabling efficient offline training and fast online QoI evaluation. An adaptive training set further improves convergence and mitigates overfitting, with numerical results across 1D and 2D problems showing high QoI accuracy on coarse meshes and substantial computational efficiency gains.

Abstract

The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a parametric PDE. The central component in our approach is an efficient neural-network-weighted Minimal-Residual formulation, which, after training, provides Galerkin-based approximations in standard discrete spaces that have accurate quantities of interest, regardless of the coarseness of the discrete space.
Paper Structure (20 sections, 49 equations, 9 figures, 1 table, 1 algorithm)

This paper contains 20 sections, 49 equations, 9 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Deep learning MinRes methodology
  3. The abstract parametric problem
  4. Main idea of Neural-weighted MinRes
  5. Weighted inner-products
  6. Weighted residual minimization
  7. Training of neural-weighted MinRes discrete system
  8. Exemplification
  9. Ensuring Efficiency
  10. Efficient assembly: Precompute sub-domain Gram matrices
  11. Efficient assembly: Precompute stiffness matrices and load vectors with unit parameters
  12. Offline and Online procedures
  13. Neural-Network Adaptive Training Set
  14. Adaptive generation of training data
  15. Numerical Results
  16. ...and 5 more sections

Figures (9)

  • Figure 1: Comparison between a standard (unweighted) Galerkin method (MRes) and our trained neural-weighted-method (w-MRes) on a coarse mesh consisting of one linear finite element. The analytical solution is also shown. Note the accuracy of w-MRes at the quantity of interest (QoI), i.e., the value of $u$ at $x_0=0.7$.
  • Figure 2: Supervised training process for the artificial neural network and the inner-product weight.
  • Figure 3: Adaptive training and validation sets.
  • Figure 4: ML-MinRes performance for 1D parametric diffusion reaction example.
  • Figure 5: ML-MinRes performance for two-parameters diffusion reaction example. Red dots show the training points.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Example 2.1
  • Remark 2.1
  • Remark 3.1
  • Example 3.1
  • Remark 3.2: Practical positive piecewise polynomial weights
  • Remark 4.1