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A non-intrusive neural-network based BFGS algorithm for parameter estimation in non-stationary elasticity

Stefan Frei, Jan Reichle, Stefan Volkwein

TL;DR

The paper tackles parameter estimation under expensive PDE constraints in non-stationary elasticity. It introduces a non-intrusive optimization framework that replaces the PDE solver with an offline-trained neural-network surrogate of the FE solver, enabling gradient- and BFGS-based estimation of elasticity parameters E and nu from measurements. On a dynamic elastic contact problem, the NN surrogate achieves close fidelity to the FE objective and enables fast convergence (e.g., 7 iterations for BFGS) with small parameter errors. The approach promises substantial computational savings in PDE-constrained inverse problems when the forward model is expensive or lacks accessible sensitivities.

Abstract

We present a non-intrusive gradient and a non-intrusive BFGS algorithm for parameter estimation problems in non-stationary elasticity. To avoid multiple (and potentially expensive) solutions of the underlying partial differential equation (PDE), we approximate the PDE solver by a neural network within the algorithms. The network is trained offline for a given set of parameters. The algorithms are applied to an unsteady linear-elastic contact problem; their convergence and approximation properties are investigated numerically.

A non-intrusive neural-network based BFGS algorithm for parameter estimation in non-stationary elasticity

TL;DR

The paper tackles parameter estimation under expensive PDE constraints in non-stationary elasticity. It introduces a non-intrusive optimization framework that replaces the PDE solver with an offline-trained neural-network surrogate of the FE solver, enabling gradient- and BFGS-based estimation of elasticity parameters E and nu from measurements. On a dynamic elastic contact problem, the NN surrogate achieves close fidelity to the FE objective and enables fast convergence (e.g., 7 iterations for BFGS) with small parameter errors. The approach promises substantial computational savings in PDE-constrained inverse problems when the forward model is expensive or lacks accessible sensitivities.

Abstract

We present a non-intrusive gradient and a non-intrusive BFGS algorithm for parameter estimation problems in non-stationary elasticity. To avoid multiple (and potentially expensive) solutions of the underlying partial differential equation (PDE), we approximate the PDE solver by a neural network within the algorithms. The network is trained offline for a given set of parameters. The algorithms are applied to an unsteady linear-elastic contact problem; their convergence and approximation properties are investigated numerically.
Paper Structure (7 sections, 1 theorem, 15 equations, 7 figures, 2 algorithms)

This paper contains 7 sections, 1 theorem, 15 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Finite Element (FE) Discretization
  4. Neural Network Approximation
  5. Neural-Network Based Gradient and BFGS Algorithms
  6. Numerical Examples
  7. Conclusion

Key Result

Lemma 5.1

Let $N(W,B;p)$ be the dense network defined in Section Section:3. Moreover, let $z^{l}=W^{l}a^{l-1}+b^{l}$ and $a^{l}=\sigma_l(z^{l})$ be the output of the $l$-the layer of the network. It holds for the derivatives $\rho^l=(\rho^{l}_j)_{j=1}^{n_l}$, $\rho^{l}_j=\partial_{a^{l}_j}\mathcal{F}^N$: where $(u\circ v)_j=u_jv_j$. In particular, we obtain the derivative $\rho_j^{0}\colonequals\partial_{p

Figures (7)

  • Figure 1: Spatial domain $\Omega$ and its finite element triangulation $\mathscr T_h$.
  • Figure 2: Dense neural network with two hidden layers. We use the least-squares cost functional $\mathcal{L}(p^{(i)})=\frac{1}{2n}\sum_{i=1}^n{\|C^{(i)}-N(p^{(i)}) \|}_{{\mathbb{R}}^m}^2$ for given parameters $p^i=(E^i, \nu^i)$ and corresponding observations $C^{(i)}=C(u_i^h),\, i=1,\ldots,n$.
  • Figure 3: Comparison of the FE objective $\mathcal{F}^h(p_1)$ and its neural network approximation $\mathcal{F}^N(p_1)$ for ${{\bm u}^\mathrm{obs}}=\mathcal{C}({\bm u}_1^h)$ and ${\bm u}^h_1=\mathcal{S}^h(p_1)$ with $p_1$ defined in \ref{['punkt1']}.
  • Figure 4: Convergence behavior of the neural-network based gradient algorithm applied to problem \ref{['ges_min_funk']} with exact solutions $p_1$\ref{['punkt1']} (left sketch) and $p_2$ (right sketch), respectively. The correct solutions are marked with blue dots, the iterates in red.
  • Figure 5: Left: Objective function ${\cal F}^N$, right: Norm of the gradient $\|\nabla_p {\cal F}^N\|_{{\mathbb{R}}^2}$ for the gradient algorithm applied to the first example corresponding to parameters \ref{['punkt1']}.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Lemma 5.1