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Robust error estimates of PINN in one-dimensional boundary value problems for linear elliptic equations

Jihahm Yoo, Haesung Lee

TL;DR

This paper rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach and establishes robust error estimates of PINN regardless of the quantities of the coefficients.

Abstract

In this paper, we study physics-informed neural networks (PINN) to approximate solutions to one-dimensional boundary value problems for linear elliptic equations and establish robust error estimates of PINN regardless of the quantities of the coefficients. In particular, we rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach. Deriving $L^2$-contraction estimates, we show that the error, defined as the mean square of the differences between the true solution and our trial function at the sample points, is dominated by the training loss. Furthermore, we show that as the quantities of the coefficients for the differential equation increase, the error-to-loss ratio rapidly decreases. Our theoretical and experimental results confirm the robustness of the error regardless of the quantities of the coefficients.

Robust error estimates of PINN in one-dimensional boundary value problems for linear elliptic equations

TL;DR

This paper rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach and establishes robust error estimates of PINN regardless of the quantities of the coefficients.

Abstract

In this paper, we study physics-informed neural networks (PINN) to approximate solutions to one-dimensional boundary value problems for linear elliptic equations and establish robust error estimates of PINN regardless of the quantities of the coefficients. In particular, we rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach. Deriving -contraction estimates, we show that the error, defined as the mean square of the differences between the true solution and our trial function at the sample points, is dominated by the training loss. Furthermore, we show that as the quantities of the coefficients for the differential equation increase, the error-to-loss ratio rapidly decreases. Our theoretical and experimental results confirm the robustness of the error regardless of the quantities of the coefficients.
Paper Structure (16 sections, 17 theorems, 79 equations, 4 figures)

This paper contains 16 sections, 17 theorems, 79 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. Idea of the error estimates in PINN
  4. Main results
  5. Structure of this paper
  6. Notations and basic results
  7. Energy estimates
  8. Existence and uniqueness with energy estimates
  9. Numerical experiments for PINN with energy estimates
  10. Error analysis via $L^2(I, \mu)$-contraction estimates
  11. $L^2(I, \mu)$-contraction estimates
  12. Numerical experiments for PINN with $L^2(I, \mu)$-contraction estimates
  13. Error analysis via $L^2(I, dx)$-contraction estimates
  14. $L^2(I, dx)$-contraction estimates
  15. Numerical experiments for PINN with $L^2(I, dx)$-contraction estimates
  16. ...and 1 more sections

Key Result

Proposition 2.1

Let $u \in H^{1,p}(I)$, where $p \in [1, \infty]$ and $I$ is a (possibly unbounded) open interval. Then, the following hold:

Figures (4)

  • Figure 1: Not effective (PINN I)
  • Figure 2: Successful (PINN II)
  • Figure 3: Comparison of Relative error between two methods as epoch increases
  • Figure 4: As $\lambda$ increases, Loss increases but Error remains robust due to the decrease of $\frac{\sf Error}{\sf Loss}$

Theorems & Definitions (23)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3: Energy estimates
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Example 3.7
  • ...and 13 more