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Abstract When performing time-intensive optimization tasks, such as those in topology or shape optimization, researchers have turned to machine-learned inverse design (ID) methods—i.e., predicting the optimized geometry from input conditions—to replace or warm start traditional optimizers. Such methods are often optimized to reduce the mean squared error (MSE) or binary cross entropy between the output and a training dataset of optimized designs. While convenient, we show that this choice may be myopic. Specifically, we compare two methods of optimizing the hyperparameters of easily reproducible machine learning models including random forest, k-nearest neighbors, and deconvolutional neural network model for predicting the three optimal topology problems. We show that under both direct inverse design and when warm starting further topology optimization, using MSE metrics to tune hyperparameters produces less performance models than directly evaluating the objective function, though both produce designs that are almost one order of magnitude better than using the common uniform initialization. We also illustrate how warm starting impacts both the convergence time, the type of solutions obtained during optimization, and the final designs. Overall, our initial results portend that researchers may need to revisit common choices for evaluating ID methods that subtly tradeoff factors in how an ID method will actually be used. We hope our open-source dataset and evaluation environment will spur additional research in those directions. 

Abstract Adjoint-based design optimizations are usually computationally expensive and those costs scale with resolution. To address this, researchers have proposed machine learning approaches for inverse design that can predict higher-resolution solutions from lower cost/resolution ones. Due to the recent success of diffusion models over traditional generative models, we extend the use of diffusion models for multi-resolution tasks by proposing the conditional cascaded diffusion model (cCDM). Compared to GANs, cCDM is more stable to train, and each diffusion model within the cCDM can be trained independently, thus each model’s parameters can be tuned separately to maximize the performance of the pipeline. Our study compares cCDM against a cGAN model with transfer learning. Our results demonstrate that the cCDM excels in capturing finer details, preserving volume fraction constraints, and minimizing compliance errors in multi-resolution tasks when a sufficient amount of high-resolution training data (more than 102 designs) is available. Furthermore, we explore the impact of training data size on the performance of both models. While both models show decreased performance with reduced high-resolution training data, the cCDM loses its superiority to the cGAN model with transfer learning when training data is limited (less than 102), and we show the break-even point for this transition. Also, we highlight that while the diffusion model may achieve better pixel-wise performance in both low-resolution and high-resolution scenarios, this does not necessarily guarantee that the model produces optimal compliance error or constraint satisfaction. 

Design optimization, and particularly adjoint-based multi-physics shape and topology optimization, is time-consuming and often requires expensive iterations to converge to desired designs. In response, researchers have developed Machine Learning (ML) approaches — often referred to as Inverse Design methods — to either replace or accelerate tools like Topology optimization (TO). However, these methods have their own hidden, non-trivial costs including that of data generation, training, and refinement of ML-produced designs. This begs the question: when is it actually worth learning Inverse Design, compared to just optimizing designs without ML assistance? This paper quantitatively addresses this question by comparing the costs and benefits of three different Inverse Design ML model families on a Topology Optimization (TO) task, compared to just running the optimizer by itself. We explore the relationship between the size of training data and the predictive power of each ML model, as well as the computational and training costs of the models and the extent to which they accelerate or hinder TO convergence. The results demonstrate that simpler models, such as K-Nearest Neighbors and Random Forests, are more effective for TO warmstarting with limited training data, while more complex models, such as Deconvolutional Neural Networks, are preferable with more data. We also emphasize the need to balance the benefits of using larger training sets with the costs of data generation when selecting the appropriate ID model. Finally, the paper addresses some challenges that arise when using ML predictions to warmstart optimization, and provides some suggestions for budget and resource management.