Good Enough is Better: Feasibility vs. Pareto-Optimality in Alloy Design

TL;DR

The paper benchmarks constraint-satisfaction versus Pareto-optimality driven alloy design under multiple hard constraints. It leverages Gaussian Process Classifiers for feasibility, Gaussian Process Regressors for objectives, and metrics such as the probability of feasibility $PoF$, the time-to-first-feasible $TTFF$, and the hypervolume $HV$ to compare campaigns, introducing a constraint-first paradigm. Results show constraint-satisfaction identifies feasible alloys faster and more reliably than optimization, with informative priors further reducing $TTFF$. The authors advocate a hybrid workflow that first secures feasibility and then expands the Pareto frontier, offering a decision-relevant path for industrial materials discovery, especially in highly constrained design spaces.

Abstract

In alloy design, the search for candidate materials is often framed as an optimization problem, with the goal of identifying Pareto-optimal solutions across multiple objectives. However, Pareto-optimal solutions do not necessarily satisfy all minimum performance thresholds required for practical deployment. An alternative approach is to treat alloy design as a constraint satisfaction problem, in which the goal is to identify any solution that meets all bare minimum requirements across multiple quantities of interest. These approaches have yet to be benchmarked against each other in the context of realistic alloy design problems. In this work, we demonstrate that, in realistic alloy design campaigns involving multiple objectives and constraints, the constraint satisfaction framework yields a higher likelihood of finding viable alloys than optimization-based approaches. Furthermore, constraint-satisfaction approaches find the first viable alloy solutions earlier than optimization. Our results suggest that focusing on feasibility rather than optimality can lead to more actionable outcomes in materials discovery, particularly in highly constrained applications.

Figures (8)

• Figure 1: Categorical Gaussian Processes (GPCs) are used to model binary feasibility constraints during both constraint satisfaction and constrained optimization. Latent GP $f(x)$ fit to binary labels. Feasibility probability $p(y{=}1\mid x)=\sigma\!(f(x))$ with the same uncertainty propagated via $\sigma(\mu_f \pm \sigma_f)$. This approach is appropriate for modeling single-phase stability, as demonstrated previously hardcastle2025physics, and is amenable to incorporating physics-based prior knowledge via informative prior mean functions to accelerate learning; see hardcastle2025physics for details.
• Figure 2: Classification of continuous properties using a GPR. The figure on the left shows a 1D regression problem, where the black dots represent the data used to train the model and the dashed line represents an objective constraint ($y\ge5$). The mean prediction $\mu(x)$ and standard deviation $\pm \sigma(x)$ is highlighted for the arbitrary input $x=11$, and its associated posterior predictive distribution is shown on the righthand figure. Since the posterior predictive distribution is a normal distribution, the CDF can be used to dertermine the probability of meeting the constraint.
• Figure 3: PoF calculations across the design space. The Nb–Mo–Ta–V–W–Cr system is represented using a berycentric projection. The markers show the data used to train the GP. After training, the probability of meeting each invdividual constraint is computed across the design space (upper five projections). Assuming conditional independence, the PoF is the product of the individual probabilities (bottom projection). Note, overlap occurs on berycentric projections when datapoints are mapped closely together. This limitation is discussed in our previous work vela2025visualizing.
• Figure 4: Bi-objective hypervolume (minimization). The shaded staircase is the region dominated by the Pareto set $\mathcal{P}$ and bounded by the reference point $\mathbf r$.
• Figure 5: Closed-loop constraint–satisfaction workflow to minimize time-to-first-feasible (TTFF).