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Quasi-Optimal Least Squares: Inhomogeneous boundary conditions, and application with machine learning

Harald Monsuur, Robin Smeets, Rob Stevenson

TL;DR

The paper develops a Quasi-Optimal Least Squares framework for PDEs with inhomogeneous boundary conditions, formulating residual minimization in evaluable norms and replacing non-evaluable boundary norms with discretized dual norms to preserve quasi-best approximations. It provides a rigorous inf-sup stable construction, including Fortin-type interpolators and Riesz lifts, and extends the theory to Stokes equations and neural network discretizations via a minimax (adversarial) approach. By avoiding fractional Sobolev norms on the boundary, the method enables practical finite-element and machine-learning implementations that maintain quasi-optimal convergence, even for problems with singular solutions. Numerical results demonstrate adaptive FE performance and competitive ML performance, illustrating the method’s potential to improve boundary-condition handling in PDE solvers while highlighting remaining challenges in quadrature and non-convex optimization.

Abstract

We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a quasi-best approximations from the employed trial spaces. Dual norms do enter the least-squares functional, so that solving the least squares problem amounts to solving a saddle point or minimax problem. For finite element applications we construct uniformly stable finite element pairs, whereas for Machine Learning applications we employ adversarial networks.

Quasi-Optimal Least Squares: Inhomogeneous boundary conditions, and application with machine learning

TL;DR

The paper develops a Quasi-Optimal Least Squares framework for PDEs with inhomogeneous boundary conditions, formulating residual minimization in evaluable norms and replacing non-evaluable boundary norms with discretized dual norms to preserve quasi-best approximations. It provides a rigorous inf-sup stable construction, including Fortin-type interpolators and Riesz lifts, and extends the theory to Stokes equations and neural network discretizations via a minimax (adversarial) approach. By avoiding fractional Sobolev norms on the boundary, the method enables practical finite-element and machine-learning implementations that maintain quasi-optimal convergence, even for problems with singular solutions. Numerical results demonstrate adaptive FE performance and competitive ML performance, illustrating the method’s potential to improve boundary-condition handling in PDE solvers while highlighting remaining challenges in quadrature and non-convex optimization.

Abstract

We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a quasi-best approximations from the employed trial spaces. Dual norms do enter the least-squares functional, so that solving the least squares problem amounts to solving a saddle point or minimax problem. For finite element applications we construct uniformly stable finite element pairs, whereas for Machine Learning applications we employ adversarial networks.

Paper Structure

This paper contains 29 sections, 14 theorems, 129 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Approach from 204.19
  3. Current work
  4. Application with Machine Learning
  5. Organization
  6. Notations
  7. Least squares principles in an abstract setting
  8. Continuous problem
  9. Discretization
  10. Implementation
  11. A posteriori error estimation
  12. Application
  13. Model elliptic second order boundary value problem
  14. Well-posed first order system reformulations
  15. Avoidance of fractional Sobolev norms
  16. ...and 14 more sections

Key Result

Theorem 2.1

Assuming that $G_1 X^\delta \neq \{0\}$ and $Y_1^\delta \neq \{0\}$, let Then $\gamma^\delta \geq \|\Pi^\delta\|_{\mathcal{L}(Y_1,Y_1)}^{-1}$. Conversely, if $\gamma^\delta >0$, then there exists a $\Pi^\delta$ as in fortin, being even a projector onto $Y_1^\delta$, with $\|\Pi^\delta\|^{-1}_{\mathcal{L}(Y_1,Y_1)} = \gamma^\delta$.

Figures (4)

  • Figure 1: Results with Sect. \ref{['sec:numfem']} (FEM). Top row: #DoFs in $P^\delta\times U^\delta$ vs. $\mathcal{E}(\vec{\lambda}_{\rm{\ref{['item:b']}}}^\delta, \lambda_{\rm{\ref{['item:c']}}}^\delta, u^\delta, p^\delta)$ for $q=0$ (left) and $q=1$ (right). Bottom left: #DoFs in $P^\delta\times U^\delta$ vs. effectivity index $\mathcal{E}(\vec{\lambda}_{\rm{\ref{['item:b']}}}^\delta, \lambda_{\rm{\ref{['item:c']}}}^\delta, u^\delta, p^\delta)/ \sqrt{\|\nabla u -p^\delta\|_{H(\mathop{\mathrm{div}}\nolimits;\Omega)}^2+\|u-u^\delta\|_{H^1(\Omega)}^2}$. Bottom right: #DoFs in $P^\delta\times U^\delta$ vs. different parts of the error estimator $\mathcal{E}(\vec{\lambda}_{\rm{\ref{['item:b']}}}^\delta, \lambda_{\rm{\ref{['item:c']}}}^\delta, u^\delta, p^\delta)$, all multiplied with $1/\sqrt{\|\nabla u -p^\delta\|_{H(\mathop{\mathrm{div}}\nolimits;\Omega)}^2+\|u-u^\delta\|_{H^1(\Omega)}^2}$, for adaptive refinement with $q=0$.
  • Figure 2: Results with Sect. \ref{['sec:numfem']} (FEM). Meshes ${\mathcal{T}}^\delta$, ${\mathcal{T}}_D^\delta$ and ${\mathcal{T}}_N^\delta$, where $\Omega$ has been rotated over $90^\circ$ counterclockwise.
  • Figure 3: Graphical representation Residual Neural Network (ResNet).
  • Figure 4: Illustration with Ex. \ref{['example:1']} (Machine Learning). Plot of the $H^1(\Omega)$-error $\|u^\ast - u_\theta\|^2_{H^1(\Omega)}$ against epoch for the different methods, using adaptive quadrature integration.

Theorems & Definitions (31)

  • remark 1.1
  • Theorem 2.1: 249.992
  • Theorem 2.2
  • proof
  • Theorem 2.3: 204.19
  • remark 2.4
  • Proposition 2.5: 204.19
  • remark 2.6
  • remark 2.7
  • remark 3.1
  • ...and 21 more