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A neural network approach to learning solutions of a class of elliptic variational inequalities

Amal Alphonse, Michael Hintermüller, Alexander Kister, Chin Hang Lun, Clemens Sirotenko

TL;DR

A weak adversarial approach to solving obstacle problems using neural networks, which is able to easily handle obstacle problems that feature biactivity (or lack of strict complementarity), a situation that poses difficulty for traditional numerical methods.

Abstract

We develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational inequality) as a minmax problem, providing a natural formulation amenable to learning. Our approach, in contrast to much of the literature, does not require the elliptic operator to be symmetric. We provide an error analysis for suitable discretisations of the continuous problem, estimating in particular the approximation and statistical errors. Parametrising the solution and test function as neural networks, we apply a modified gradient descent ascent algorithm to treat the problem and conclude the paper with various examples and experiments. Our solution algorithm is in particular able to easily handle obstacle problems that feature biactivity (or lack of strict complementarity), a situation that poses difficulty for traditional numerical methods.

A neural network approach to learning solutions of a class of elliptic variational inequalities

TL;DR

A weak adversarial approach to solving obstacle problems using neural networks, which is able to easily handle obstacle problems that feature biactivity (or lack of strict complementarity), a situation that poses difficulty for traditional numerical methods.

Abstract

We develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational inequality) as a minmax problem, providing a natural formulation amenable to learning. Our approach, in contrast to much of the literature, does not require the elliptic operator to be symmetric. We provide an error analysis for suitable discretisations of the continuous problem, estimating in particular the approximation and statistical errors. Parametrising the solution and test function as neural networks, we apply a modified gradient descent ascent algorithm to treat the problem and conclude the paper with various examples and experiments. Our solution algorithm is in particular able to easily handle obstacle problems that feature biactivity (or lack of strict complementarity), a situation that poses difficulty for traditional numerical methods.

Paper Structure

This paper contains 34 sections, 20 theorems, 174 equations, 15 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Analysis of the continuous problem
  3. Basic properties and saddle points
  4. Minmax approach via the regularised gap function
  5. Relaxations of the problem
  6. Imposition of the boundary conditions
  7. Full penalty approach
  8. Generalised approach
  9. Neural network approach
  10. Neural networks with homogeneous BC.
  11. Discretised problems
  12. Imposition of the boundary conditions
  13. Full penalty approach
  14. Generalised approach
  15. Remarks on existence
  16. ...and 19 more sections

Key Result

Proposition 2.1

Let $A:= -\Delta + \sum_{i=1}^n b_i\partial_{x_i} + c\mathrm{Id}$ with coefficients $b_i, c \in L^\infty(\Omega)$, $c \geq c_0 \geq 0$ for a constant $c_0$, such that the coercivity condition eq:conditions_on_A_2 is satisfied, $f \in L^2(\Omega)$, $h \in H^{3\slash 2}(\partial\Omega)$, $\psi \in H^2

Figures (15)

  • Figure 1: Example 1. The difference between the learned solution $u_{\mathrm{NN}}$ and the true solution $u_{\mathrm{exact}}$ is smaller than 0.02. On the coincidence set $[{1}\slash {2\sqrt{2}}, 1-{1}\slash {2\sqrt{2}}]\approx [0.354,0.646]$, $u_{\mathrm{NN}}$ violates the obstacle condition nearly constantly. The maximal difference is comparable to the one obtained in DeepNeural.
  • Figure 2: Training trajectories for Example 1.
  • Figure 3: The non-symmetric Example 2. The difference has a magnitude smaller than $0.007$.
  • Figure 4: Training trajectories for Example 2. Compared to \ref{['fig:L2_Linf_Errors_1D_eg']}, the convergence happens earlier.
  • Figure 5: Mean (over 50 seeds) of $L^2$ and $H^1$ errors for Example 2 for different values of $w_{o_1}=w_{o_2}$.
  • ...and 10 more figures

Theorems & Definitions (47)

  • Proposition 2.1: $H^2$-regularity
  • proof
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 37 more