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Automatic differentiation (AD) was introduced into topology optimization (TO) more than two decades ago to compute accurate gradients through complex computational workflows. Nevertheless, its adoption within the TO community has remained limited, largely due to the strong reliance on adjoint-based sensitivity analysis—which typically offers superior memory efficiency and runtime performance—and the practical difficulties of integrating large-scale simulations into specialized AD frameworks. The recent rise of machine learning (ML) has opened new opportunities for TO through the advanced AD capabilities of modern ML frameworks such as JAX and PyTorch. A growing body of work at the intersection of ML and TO now focuses on tightly coupling ML components with classical TO workflows. Neural TO is a prominent example, in which an untrained neural network parameterizes the material density field and optimization proceeds over the network parameters. To enable such ML–TO hybrid workflows, a deeper understanding of how AD systems operate in these frameworks is essential. This article explains the practical principles of AD in modern ML frameworks and their relation to classical adjoint-based sensitivity analysis. We present implementation strategies for wrapping essential operations—such as finite element solvers—into AD-compatible components without reimplementing them from scratch. These ideas are illustrated through two compact code examples: a classical TO pipeline with selectively AD-wrapped components and a neural TO workflow.
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Automatic differentiation (AD) was introduced into topology optimization (TO) more than two decades ago to compute accurate gradients through complex computational workflows. Nevertheless, its adoption within the TO community has remained limited, largely due to the strong reliance on adjoint-based sensitivity analysis—which typically offers superior memory efficiency and runtime performance—and the practical difficulties of integrating large-scale simulations into specialized AD frameworks. The recent rise of machine learning (ML) has opened new opportunities for TO through the advanced AD capabilities of modern ML frameworks such as JAX and PyTorch. A growing body of work at the intersection of ML and TO now focuses on tightly coupling ML components with classical TO workflows. Neural TO is a prominent example, in which an untrained neural network parameterizes the material density field and optimization proceeds over the network parameters. To enable such ML–TO hybrid workflows, a deeper understanding of how AD systems operate in these frameworks is essential. This article explains the practical principles of AD in modern ML frameworks and their relation to classical adjoint-based sensitivity analysis. We present implementation strategies for wrapping essential operations—such as finite element solvers—into AD-compatible components without reimplementing them from scratch. These ideas are illustrated through two compact code examples: a classical TO pipeline with selectively AD-wrapped components and a neural TO workflow.
Neural networks (NNs) hold great promise for advancing inverse design via topology optimization (TO), yet misconceptions about their application persist. This article focuses on neural topology optimization (neural TO), which leverages NNs to reparameterize the decision space and reshape the optimization landscape. While the method is still in its infancy, our analysis tools reveal critical insights into the NNs’ impact on the optimization process. We demonstrate that the choice of NN architecture significantly influences the objective landscape and the optimizer’s path to an optimum. Notably, NNs introduce non-convexities even in otherwise convex landscapes, potentially delaying convergence in convex problems but enhancing exploration for non-convex problems. This analysis lays the groundwork for future advancements by highlighting: (1) the potential of neural TO for non-convex problems and dedicated GPU hardware (the “good”), (2) the limitations in smooth landscapes (the “bad”), and (3) the complex challenge of selecting optimal NN architectures and hyperparameters for superior performance (the “ugly”).
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Neural networks (NNs) hold great promise for advancing inverse design via topology optimization (TO), yet misconceptions about their application persist. This article focuses on neural topology optimization (neural TO), which leverages NNs to reparameterize the decision space and reshape the optimization landscape. While the method is still in its infancy, our analysis tools reveal critical insights into the NNs’ impact on the optimization process. We demonstrate that the choice of NN architecture significantly influences the objective landscape and the optimizer’s path to an optimum. Notably, NNs introduce non-convexities even in otherwise convex landscapes, potentially delaying convergence in convex problems but enhancing exploration for non-convex problems. This analysis lays the groundwork for future advancements by highlighting: (1) the potential of neural TO for non-convex problems and dedicated GPU hardware (the “good”), (2) the limitations in smooth landscapes (the “bad”), and (3) the complex challenge of selecting optimal NN architectures and hyperparameters for superior performance (the “ugly”).
