Neural Networks for Generating Better Local Optima in Topology Optimization

TL;DR

This work investigates whether neural-network material discretizations can yield better local optima in acoustic topology optimization, addressing a challenging forward problem with potential pitfalls in local convergence. It compares a conventional linear ansatz to a neural-network ansatz (NN) for predicting design variables, and introduces transfer-learning-based schemes to mitigate bypassing filtering/projection and to seed the NN with favorable initializations. The results show that a pretrained NN, particularly when combined with transfer learning, can identify notably better optima across multiple frequencies, at the cost of higher computational effort and with results that are highly problem-dependent. The study highlights that NN discretizations can outperform traditional methods in unconstrained first-order optimization, while constrained or higher-order optimization remains challenging and would require further methodological development for broad applicability.

Abstract

Neural networks have recently been employed as material discretizations within adjoint optimization frameworks for inverse problems and topology optimization. While advantageous regularization effects and better optima have been found for some inverse problems, the benefit for topology optimization has been limited -- where the focus of investigations has been the compliance problem. We demonstrate how neural network material discretizations can, under certain conditions, find better local optima in more challenging optimization problems, where we here specifically consider acoustic topology optimization. The chances of identifying a better optimum can significantly be improved by running multiple partial optimizations with different neural network initializations. Furthermore, we show that the neural network material discretization's advantage comes from the interplay with the Adam optimizer and emphasize its current limitations when competing with constrained and higher-order optimization techniques. At the moment, this discretization has only been shown to be beneficial for unconstrained first-order optimization.

Figures (20)

• Figure 1: Topology optimization of a ceiling $\Omega_c$ of height $h_c$ in order to suppress the sound pressure in the domain $\Omega_s$ from a harmonic source $s$ at $x_f, y_f$ with frequency $f$. The suppressed domain is rectangular with dimensions $a_s, b_s$ and centered at $x_s, y_s$. The domain $\Omega$ with dimensions $a\times b$, including $\Omega_c$ and $\Omega_s$, is governed by the acoustic wave equation.
• Figure 2: Sound pressure level distribution in dB at $f=69.43$ Hz without and with an optimized ceiling
• Figure 3: Rosenbrock function optimized with NN ansatz and directly on the coordinates (linear ansatz) using Adam
• Figure 4: Rosenbrock function optimized with NN ansatz and directly on the coordinates (linear ansatz) using steepest descent
• Figure 5: Optimized structure after 300 epochs (280 with $q=2, n_v=4$ and 20 with $q=4, n_v=4$) with the corresponding sound pressure level $L_p$ in the to-be-suppressed area