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A discontinuous plane wave neural network method for Helmholtz equation and time-harmonic Maxwell's equations

Long Yuan, Qiya Hu

TL;DR

This DPWNN method can generate approximate solutions with higher accuracy than the PWLS method and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer.

Abstract

In this paper we propose a {\it discontinuous} plane wave neural network (DPWNN) method with $hp-$refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with $h-$refinement and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer, where the activation function on each element is chosen as a complex-valued exponential function like the plane wave function. The desired approximate solution is recursively generated by iteratively solving the minimization problem associated with the functional and the sets described above, which is defined by a sequence of approximate minimizers of the underlying residual functionals, where plane wave direction angles and activation coefficients are alternatively computed by iterative algorithms. For the proposed DPWNN method, the plane wave directions are adaptively determined in the iterative process, which is different from that in the standard PWLS method (where the plane wave directions are preliminarily given). Numerical experiments will confirm that this DPWNN method can generate approximate solutions with higher accuracy than the PWLS method.

A discontinuous plane wave neural network method for Helmholtz equation and time-harmonic Maxwell's equations

TL;DR

This DPWNN method can generate approximate solutions with higher accuracy than the PWLS method and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer.

Abstract

In this paper we propose a {\it discontinuous} plane wave neural network (DPWNN) method with refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with refinement and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer, where the activation function on each element is chosen as a complex-valued exponential function like the plane wave function. The desired approximate solution is recursively generated by iteratively solving the minimization problem associated with the functional and the sets described above, which is defined by a sequence of approximate minimizers of the underlying residual functionals, where plane wave direction angles and activation coefficients are alternatively computed by iterative algorithms. For the proposed DPWNN method, the plane wave directions are adaptively determined in the iterative process, which is different from that in the standard PWLS method (where the plane wave directions are preliminarily given). Numerical experiments will confirm that this DPWNN method can generate approximate solutions with higher accuracy than the PWLS method.
Paper Structure (21 sections, 5 theorems, 88 equations, 11 figures, 1 algorithm)

This paper contains 21 sections, 5 theorems, 88 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. A minimization method for Helmholtz equations
  3. Discontinuous plane wave neural networks
  4. A universal approximation
  5. Pre-definition of plane wave propagation directions
  6. A discontinuous plane wave neural network iterative method
  7. Main results
  8. Discussions on Algorithm 4.1
  9. Calculation of the coefficients $c^l$
  10. Calculation of the direction angles
  11. A DPWNN method for Maxwell's equations
  12. A minimization problem
  13. Discontinuous plane wave neural networks
  14. A discontinuous plane wave neural network method
  15. Numerical experiments
  16. ...and 6 more sections

Key Result

Lemma 3.1

\newlabelelementappro Let $2\leq s\leq {m+1\over 2}$ with a sufficiently large $m$. Let $u\in H^{s}(\Omega_k)$ be a solution of the homogeneous Helmholtz equation for each element $\Omega_k$. Then, there exists $[c_1^{(k)},c_2^{(k)},\cdots,c_n^{(k)}] \in \mathbb{C}^n$ and $[{\bf d}_1^{(k)}, {\bf d} where $\lambda>0$ is a constant depending only on the shape of the elements (in particular, $\lambda

Figures (11)

  • Figure 6.1: Displacement of a string (Helmholtz equation in two dimensions). (Left) Errors at each quasi-minimization iteration. (Right) The progress of the loss function within each quasi-minimization iteration.
  • Figure 6.2: The trained set $\Phi^r$ in the lower-left element defining the final propagation angles of $\xi_r^{\theta}$ at several stages for $r = 1, 3, 5$.
  • Figure 6.3: Comparison between the DPWNN and the PWLS method (Helmholtz equation in two dimensions). (Left) Errors at each quasi-minimization iteration. (Right) Proportion of computing time w.r.t. $\omega$.
  • Figure 6.4: Displacement of a string (Helmholtz equation in three dimensions). (Left) Errors at each quasi-minimization iteration. (Right) The progress of the loss function within each quasi-minimization iteration.
  • Figure 6.5: Displacement of a membrane (Helmholtz equation in three dimensions). Exact error $|u-u_{i-1}|$ for i = 4, 6.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Lemma 3.1
  • Theorem 3.2
  • Remark 3.1
  • Definition 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Remark 4.1
  • Remark 4.2
  • Remark 5.1
  • Theorem 5.1