Two-Stage Surrogate Modeling for Data-Driven Design Optimization with Application to Composite Microstructure Generation

TL;DR

The paper tackles inverse design challenges in engineering by introducing a two-stage surrogate framework that couples a learner (to reduce the search space) with an evaluator (to impose predictive uncertainty via conformal inference). The method yields prediction intervals ensuring target outcomes fall within credible ranges, improving reliability over single-stage approaches. Demonstrations on the Ishigami benchmark and a fiber-reinforced composite microstructure generator show reduced solution variance and safer, data-driven regularization across one- and two-target scenarios. The approach is versatile and broadly applicable to design optimization tasks where uncertainty quantification is critical, enabling robust interactions between heterogeneous surrogates.

Abstract

This paper introduces a novel two-stage machine learning-based surrogate modeling framework to address inverse problems in scientific and engineering fields. In the first stage of the proposed framework, a machine learning model termed the "learner" identifies a limited set of candidates within the input design space whose predicted outputs closely align with desired outcomes. Subsequently, in the second stage, a separate surrogate model, functioning as an "evaluator," is employed to assess the reduced candidate space generated in the first stage. This evaluation process eliminates inaccurate and uncertain solutions, guided by a user-defined coverage level. The framework's distinctive contribution is the integration of conformal inference, providing a versatile and efficient approach that can be widely applicable. To demonstrate the effectiveness of the proposed framework compared to conventional single-stage inverse problems, we conduct several benchmark tests and investigate an engineering application focused on the micromechanical modeling of fiber-reinforced composites. The results affirm the superiority of our proposed framework, as it consistently produces more reliable solutions. Therefore, the introduced framework offers a unique perspective on fostering interactions between machine learning-based surrogate models in real-world applications.

Figures (11)

• Figure 1: Illustrating the proposed data-driven design optimization framework using two surrogate models, which we refer to them as learner and evaluator models. The learner model $\hat{f}$ serves as a proxy for the forward model to narrow down the original search space by finding a small subset of eligible solutions $\Omega_r\subset\Omega$. The evaluator agent $\hat{\mu}$, which is distinct from the learner, uses conformal prediction to obtain prediction intervals for each member of the reduced search space $\Omega_r$ to verify that the target output falls within the constructed interval. Hence, this framework provides safety measures when there is a discrepancy between the two machine learning-based surrogates.
• Figure 2: Illustrating the shortcomings of single-stage surrogate-assisted inverse problems using a synthetic data set with two values of regularization strength: (a) $\gamma=0$ and (b) $\gamma=1$. This experiment suggests that the optimization problem is oblivious to possible inaccuracies of the surrogate model $\hat{f}$. Thus, there needs to be an effective strategy in place to reject inconsistent solutions, such as using an evaluator model to create a prediction interval, which is shown using a red vertical line. Here, both constructed prediction intervals do not contain the target output.
• Figure 3: Using the Ishigami function to generate the training data set $\mathcal{D}$ and fitted surrogate models $\hat{f}$ and $\hat{\mu}$. The learner is a nearest neighbor regressor, while we adopt a polynomial regression model as an evaluator. To measure the predictive performance of these models, we report predicted vs. true outputs on a separate test data set that was not used for training.
• Figure 4: Solving the inverse problem using the Ishigami function with a scalar target output, shown as a horizontal line, and fixed value of $x_1\in\{0.1\pi,0.2\pi,0.3\pi,0.4\pi\}$. The goal is to find the optimal configuration of $x_2$ and $x_3$ that produces the target output. We plot the mean and standard deviation of actual outputs for the solutions of the inverse problem across $20$ independent trials to capture the randomness associated with the search space $\Omega$. The proposed two-stage surrogate modeling framework is effective in terms of reducing the standard deviation and moving the average output value closer to the target output of the inverse problem.
• Figure 5: Solving the design optimization problem using the multi-output Ishigami function with a target output vector and fixed value of $x_1\in\{0.1\pi,0.2\pi,0.3\pi,0.4\pi\}$. The goal is to find the optimal configuration of $x_2$ and $x_3$ that produces the two target outputs $f_1$ and $f_2$. We plot the mean and standard deviation of actual outputs for the solutions of the inverse problem across $20$ independent trials to capture the randomness associated with the search space $\Omega$.