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A physics-informed neural network framework for modeling obstacle-related equations

Hamid El Bahja, Jan Christian Hauffen, Peter Jung, Bubacarr Bah, Issa Karambal

TL;DR

This work extends the application of physics-informed neural networks to solve obstacle-related PDEs, which are particularly challenging as they require numerical methods that can accurately approximate solutions constrained by obstacles.

Abstract

Deep learning has been highly successful in some applications. Nevertheless, its use for solving partial differential equations (PDEs) has only been of recent interest with current state-of-the-art machine learning libraries, e.g., TensorFlow or PyTorch. Physics-informed neural networks (PINNs) are an attractive tool for solving partial differential equations based on sparse and noisy data. Here extend PINNs to solve obstacle-related PDEs which present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of the solution that lies above a given obstacle. The performance of the proposed PINNs is demonstrated in multiple scenarios for linear and nonlinear PDEs subject to regular and irregular obstacles.

A physics-informed neural network framework for modeling obstacle-related equations

TL;DR

This work extends the application of physics-informed neural networks to solve obstacle-related PDEs, which are particularly challenging as they require numerical methods that can accurately approximate solutions constrained by obstacles.

Abstract

Deep learning has been highly successful in some applications. Nevertheless, its use for solving partial differential equations (PDEs) has only been of recent interest with current state-of-the-art machine learning libraries, e.g., TensorFlow or PyTorch. Physics-informed neural networks (PINNs) are an attractive tool for solving partial differential equations based on sparse and noisy data. Here extend PINNs to solve obstacle-related PDEs which present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of the solution that lies above a given obstacle. The performance of the proposed PINNs is demonstrated in multiple scenarios for linear and nonlinear PDEs subject to regular and irregular obstacles.
Paper Structure (8 sections, 32 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 8 sections, 32 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical overview
  3. Methodology
  4. Numerical results
  5. One-dimensional linear obstacle problems
  6. One-dimensional nonlinear obstacle problem
  7. Two-dimensional obstacle
  8. Conclusions and Outlook

Figures (10)

  • Figure 1: The contact set and the free boundary in the classical obstacle problem.
  • Figure 2: Neural Network Architecture for PINNs with Enforced Boundary Conditions for an Obstacle Problem.
  • Figure 3: A comparison between the minimum loss over the number of iterations for the losses defined in (3.1) and (3.2).
  • Figure 4: Impact of Network Architecture on Training Loss: (a) Layers, (b) Nodes, (c) Collocation points.
  • Figure 5: One-dimensional $\varphi_{1}$-obstacle Poisson's equation: (left) The predicted solution against the exact solution (4.4). (right) A plot of the pointwise $L^{\infty}$-error estimation.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Remark 3.1