Data-driven machine learning (ML) has made significant progress in predicting structure-property relationships, discovering new materials, and optimizing processes within materials science. However, the factors influencing materials structure-property relationships are highly complex and multifaceted, resulting in extremely intricate distribution patterns in materials data. This complexity makes it difficult for traditional data-driven approaches, which attempt to establish a single mapping between a fixed set of descriptors and target properties, to accurately and comprehensively unravel these relationships. Through the analysis of the characteristics of materials data and the modeling principle of data-driven ML methods, we have clarified the three main contradictions occurring to the application of ML in materials science, i.e., the contradictions between high dimension and small sample, accuracy and usability of models, learning results and domain knowledge. To reconcile these three contradictions, we propose the framework of machine learning embedded with materials domain knowledge and a series of novel methods. Then, by analyzing the characteristics of materials research problems and the complex and diverse structure-property relationships inherent in materials data, we believe that it is possible to effectively decompose complex material problems and address them individually by leveraging materials domain knowledge. To this end, we here propose a divide-and-conquer ML modeling strategy embedded with materials domain knowledge.

Specifically, the "Divide and Conquer" strategy aims to leverage materials domain knowledge to decompose complicated material problems and construct ML models towards each sub-problem, which can effectively reduce the complexity of ML models and ensures that the learning results of models are both accurate and credible through the integration of domain knowledge. It can be regarded as an application of the algorithmic paradigm of divide-and-conquer to ML modeling, or alternatively, as a novel form of ensemble learning. This strategy consists of: i) "Divide": Guided by materials domain knowledge, a complicated materials problem can be decomposed into several sub-problems from multiple perspectives based on an analysis of underlying physicochemical mechanisms. ii) "Conquer": Individual models are constructed for each sub-problem, where domain-specific knowledge is injected into the sub-models, establishing mappings between different ML models and distinct physicochemical mechanisms. iii) "Combine": A fusion mechanism is designed to integrate the feature extraction or decision outputs from the sub-models. As a result, multi-faceted analysis from diverse perspectives yields more comprehensive and rational insights into complex materials problems. This aids domain experts in identifying the critical thresholds and transition boundaries of the intrinsic mechanisms governing material property variations, reduces the complexity of individual ML models within each subspace, and enhances both the predictive accuracy and interpretability of the models. A further investigation reveals that, although the concept of "divide and conquer" has been widely adopted, no attempt has yet been made to theoretically derive and validate its content. In this study, we employ variational inference to provide a formal symbolic representation for the divide-and-conquer ML modeling strategy and verify its rationality and scientific validity.

Currently, researchers have begun to leverage domain knowledge to decompose materials problems or data, giving rise to diverse "divide-and-conquer" ML modeling strategies. Specific implementations include: i) Stage divide-and-conquer. Deconstructing long-term, highly complex evolutionary processes into several substages with relatively singular and stable physicochemical mechanisms, thereby simplifying the intricate. This strategy can be applied to complex materials issues featuring multi-step transition characteristics, such as the glass transition temperature (T_g) and the battery rollover point, to enable accurate prediction. ii) Positive/negative examples divide-and-conquer. Samples are categorized into distinct subspace according to their positive or negative impact on the target performance. This strategy can be applied to complex materials problems that encompass samples from superior performance to degradation, such as in the optimal design of lithium-ion battery electrolytes and the hydrothermal synthesis of inorganic-organic hybrid materials. iii) Category divide-and-conquer. By leveraging explicit labels or implicit distributional features, heterogeneous data is decomposed into several isotropic subdomains with relatively consistent physicochemical mechanisms, thereby reducing the complexity of global modeling. This strategy can be applied to the precise prediction of heterogeneous samples, such as the creep life of nickel-based single-crystal superalloys, the hardness of steel, and the strength and ductility of lead-free solder alloys. iv) Attribute divide-and-conquer. Multiple sets of optimal descriptor combinations are screened out, thereby enabling the targeted construction of several high-precision models. This strategy is validated on six distinct material systems, such as NASICON-type inorganic solid-state electrolytes, demonstrating high predictive accuracy and interpretative capabilities. v) Modality divide-and-conquer. Decomposing heterogeneous data and representing it as multi-modal data to comprehensively unravel structure-property relationships. This strategy is applicable to complex materials problems with multi-modal information, such as dual-phase steels and lithium-ion solid-state electrolytes.