Bias-Free Multiobjective Active Learning for Materials Design and Discovery

The optimal material for a given application is a central goal in materials science. With a single objective, optimization is straightforward: materials are ranked based on their performance. However, most real-world applications involve multiple, often competing, objectives. For example, in drug discovery, potency must be balanced against toxicity and solubility; in materials science, strength might conflict with ductility. In such cases, a single optimal material rarely exists, and the focus shifts to identifying the Pareto optimal set – the materials where no single objective can be improved without worsening another. This study addresses the challenge of efficiently searching for this Pareto-optimal set, particularly when evaluations are resource-intensive. Machine learning offers a powerful tool, but naive approaches like training a model on the entire design space are inefficient. Active learning provides a more effective strategy by iteratively selecting the most informative materials to evaluate, thus focusing effort on the relevant regions of the design space. This contrasts with Bayesian optimization, which typically imposes an arbitrary total order on the design space, potentially introducing bias. The proposed method avoids this bias by directly utilizing the Pareto dominance relation, a partial order, offering a more rigorous and unbiased exploration of the design space. The method’s application to *de novo* polymer design for dispersant applications, with its computationally expensive evaluations, highlights its practical utility in tackling complex materials discovery problems.

Recent research has explored the use of machine learning in materials design and discovery. Naive approaches involve training machine learning models on large datasets to predict material properties across the entire design space. However, this is computationally expensive and inefficient. Active learning offers a more efficient alternative by iteratively selecting and labeling data points that maximize the model's learning rate, thus focusing on the most informative regions of the design space. This approach is different from Bayesian optimization, commonly used for materials discovery, which implicitly assumes a total order on the search space. This total order creates a bias in the optimization process, which the proposed method avoids by directly using the Pareto dominance relation. The study highlights the limitations of conventional active learning methods, especially when dealing with incomplete datasets or noisy measurements, and proposes a solution.

The study focuses on the design of dispersants, polymers used in solid suspension systems to prevent particle flocculation. A coarse-grained model represents the polymers, simplifying the vast design space (over 53 million possible polymer sequences) for computational tractability. Three key performance indicators were selected to evaluate dispersant performance: adsorption free energy (ΔGads) onto a model surface, dimer free energy barrier (ΔGrep) representing repulsion between polymers, and radius of gyration (Rg) related to viscosity. A full factorial experimental design was used to initially sample the design space, generating 3125 unique polymer molecules. The core of the methodology is a modified version of the ε-PAL (epsilon-Pareto Active Learning) algorithm. This algorithm iteratively refines its prediction of the Pareto front by using a Gaussian process regression model to estimate both the mean and standard deviation of the objective functions for each polymer sequence. Hyperrectangles are constructed around each point, representing the uncertainty in the prediction. Using the ε-Pareto dominance relation, the algorithm confidently classifies polymers as either Pareto-optimal or discarded, focusing subsequent evaluations on the uncertain regions near the Pareto front. The algorithm selects the next polymer for evaluation by choosing the point with the largest hyperrectangle among those near or at the Pareto front. This maximizes information gain and rapidly improves the accuracy of the Pareto front approximation. The algorithm is robust to missing data, using coregionalized Gaussian process models to handle situations with incomplete datasets. After the active learning phase, the trained machine learning models are inverted using genetic algorithms (GAs) to explore the design space further and identify potential polymers not included in the initial experimental design. SHAP (SHapley Additive exPlanations) analysis provides insights into structure-property relationships, revealing which features in the polymer design most significantly affect each objective function.

The ε-PAL algorithm efficiently identifies Pareto-optimal polymers, drastically reducing the number of simulations required compared to random sampling. With an error tolerance (ε) of 0.01, the algorithm achieved the target accuracy with over 89% fewer iterations than random search (153 vs. 1421 iterations). The algorithm remains effective even with 30% missing data for one of the objective functions, highlighting its robustness to incomplete datasets. SHAP analysis provided insightful structure-property relationships, revealing how different polymer features (such as degree of polymerization and monomer type ratios) influence the key performance indicators (adsorption energy, dimer repulsion energy, and radius of gyration). The genetic algorithm used to invert the trained models did not find any polymers that Pareto-dominate the ones discovered through the active learning process, suggesting the algorithm effectively explored the relevant part of the design space. The study found that already after ten iterations the algorithm confidently discards many polymers and finds many ε-accurate Pareto-optimal polymers. The hypervolume indicator is utilized to quantitatively assess the effectiveness of the proposed approach compared to random sampling, underscoring the superior efficiency of the proposed ε-PAL algorithm. Plots show the distribution of polymer properties (radius of gyration, adsorption energy, and dimer repulsion energy) obtained from the genetic algorithm inversion and compares them to results from the original experimental design, visually demonstrating the algorithm's effectiveness. Figure 5 displays a detailed representation of polymers in property space, with sampled and Pareto-optimal points clearly marked, and projection onto respective 2D planes to visualize the Pareto front in different dimensions.

The study demonstrates the effectiveness of the bias-free multiobjective active learning approach in significantly accelerating materials discovery. The algorithm's ability to efficiently explore a vast design space and robustly handle incomplete datasets addresses limitations of conventional materials design and discovery methods. The insights gained from the SHAP analysis offer valuable understanding of the relationships between polymer structure and properties. The integration of active learning, machine learning modeling, and genetic algorithm inversion provides a comprehensive framework for materials design and discovery, emphasizing the synergy of data-driven and simulation-based approaches. The algorithm's success in finding near-optimal polymers within a fraction of the iterations needed by random search underscores its potential for various multiobjective optimization problems beyond polymer design. The successful identification of the Pareto-optimal materials, which is the most meaningful outcome in a multi-objective context, demonstrates the practical value of this approach. The results highlight that by focusing on the accuracy of the Pareto front estimation, significant gains in efficiency are achievable, compared to conventional optimization methods.

This research presents a novel bias-free multiobjective active learning algorithm that efficiently identifies Pareto-optimal materials in complex design spaces. Its application to *de novo* polymer design showcases its ability to significantly reduce the computational cost of materials discovery while providing valuable insights into structure-property relationships. Future research could explore extending the approach to even larger and more complex design spaces, incorporating diverse machine learning models and exploring different active learning strategies to further enhance the algorithm's performance and applicability in various domains.

The study used a coarse-grained model for polymers, which simplifies the complexity of real polymers. The full factorial experimental design, while effective for initial sampling, might not fully represent the entire design space. The reliance on molecular simulations for evaluation means the accuracy is dependent on the accuracy and computational cost of the simulations. The transferability of the model to different materials and experimental setups should be further investigated.