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Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equations

Beatrice Lorenz, Aras Bacho, Gitta Kutyniok

TL;DR

This paper provides rigorous error bounds for physics-informed neural networks approximating the semilinear wave equation in terms of the width of the network's layers and the number of training points for a tanh neural network with two hidden layers.

Abstract

This paper provides rigorous error bounds for physics-informed neural networks approximating the semilinear wave equation. We provide bounds for the generalization and training error in terms of the width of the network's layers and the number of training points for a tanh neural network with two hidden layers. Our main result is a bound of the total error in the $H^1([0,T];L^2(Ω))$-norm in terms of the training error and the number of training points, which can be made arbitrarily small under some assumptions. We illustrate our theoretical bounds with numerical experiments.

Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equations

TL;DR

This paper provides rigorous error bounds for physics-informed neural networks approximating the semilinear wave equation in terms of the width of the network's layers and the number of training points for a tanh neural network with two hidden layers.

Abstract

This paper provides rigorous error bounds for physics-informed neural networks approximating the semilinear wave equation. We provide bounds for the generalization and training error in terms of the width of the network's layers and the number of training points for a tanh neural network with two hidden layers. Our main result is a bound of the total error in the -norm in terms of the training error and the number of training points, which can be made arbitrarily small under some assumptions. We illustrate our theoretical bounds with numerical experiments.
Paper Structure (16 sections, 14 theorems, 85 equations, 3 figures)

This paper contains 16 sections, 14 theorems, 85 equations, 3 figures.

Table of Contents

  1. Introduction
  2. PINNs Explained
  3. Error Estimation Results for PINNs
  4. Methodology
  5. Contributions
  6. Mathematical Preliminaries
  7. The Semilinear Wave Equation
  8. Error Estimation for Semilinear Wave Equation
  9. PINN Residuals
  10. Question 1
  11. Question 2
  12. Question 3
  13. Numerical Experiments
  14. Conclusion
  15. Appendix
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $1\leq d<6$ and assume $\Omega\subset\mathbb{R}^d$ has a $C^3$ boundary. Suppose that (A1) and (A2) are fulfilled with $k\in\mathbb{N}, k>3$. Let $(u_0,u_1)\in D((-\Delta)^{\frac{k+1}{2}})\times D((-\Delta)^{\frac{k}{2}})$ where $D(-\Delta)=H^2(\Omega)\cap H^1_0(\Omega)$. Then there exists $T=T(

Figures (3)

  • Figure 1: Distribution of training points computed with the midpoint rule at the first time slice with PINN solution in the background for 144 total training points (left), 1500 total training points (middle) and 10000 total training points (right).
  • Figure 3: First row: exact solution at T=0, T=0,25 and T=0.5. Second row: PINN solution with 144 training points at T=0, T=0.25 and T=0.5. Third row: PINN solution with 18750 training points at T=0, T=0.25 and T=0.5.
  • Figure 5: The graph on the left presents the total error, training error and bound for different number of training points. In the center, the graph displays the total training error as well as individual training errors over the number of training points. On the right, the average training time per PINN in seconds over the number of training points is plotted.

Theorems & Definitions (26)

  • Definition 2.1: Neural Network DeRyck1
  • Definition 2.2: Midpoint rule DeRyck1
  • Proposition 3.1: Local Existence
  • Theorem 3.2: Global Existence
  • Corollary 3.3
  • proof
  • Theorem 4.1
  • Corollary 4.2
  • Corollary 4.3
  • proof
  • ...and 16 more