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The integration of artificial intelligence into materials science has accelerated property prediction and high-throughput screening. Yet, the field’s progress hinges on models’ ability to generalize beyond their training distributions. Existing literature often addresses generalization in broad terms, focusing on out-of-distribution performance or extrapolation without distinguishing the qualitative nature of material novelty. This conceptual manuscript introduces a novel theoretical framework for categorizing generalization in materials AI into three distinct levels: new compositions (variations within known structural families), new structures (alternative atomic arrangements or topologies), and new physics (emergence of phenomena governed by mechanisms absent from the training data). Drawing on recent advances in graph neural networks, scalable deep learning, and materials representations, we synthesize evidence that current models achieve reasonable interpolation within familiar domains but encounter progressively greater difficulties across these levels. The proposed distinction provides a structured lens for analyzing model limitations, interpreting benchmark results, and guiding the design of future architectures and training strategies. By formalizing these categories, the framework aims to advance theoretical understanding of generalization in materials AI, emphasizing the need for targeted approaches at each level to enable reliable discovery of novel materials.

Smoothness constraints have become central to the training of machine-learning interatomic potentials, whether imposed explicitly through regularization or implicitly through Sobolev objectives that couple energies with forces. Although this strategy improves interpolation, stabilizes learned potential energy surfaces, and embeds physically motivated local structure, its broader consequence is more restrictive than current practice typically acknowledges. This article shows that smoothness does not merely regularize learning within the training domain; it also limits the model’s capacity to extrapolate beyond that domain in a mathematically structured way. Working in a Sobolev-space framework, the analysis demonstrates that extrapolation error grows at least exponentially with distance from the training set, with the growth rate controlled by the force-weighting parameter λ. On that basis, the article formalizes a set of fundamental trade-offs linking interpolation accuracy to extrapolation distance, force accuracy to energy extrapolation, strong smoothness priors to representational flexibility, and data efficiency to extrapolation range. The argument further identifies the mechanisms through which these tensions arise, including prior dominance, spectral filtering, gradient-matching interference, and optimization bias toward overly smooth solutions. The result is a unified theoretical account of why Sobolev-trained potentials perform so well near known configurations yet remain fragile in genuinely novel regions of configuration space. These findings reposition smoothness from a default virtue to a task-dependent design choice and establish the need for adaptive regularization, extrapolation-aware evaluation, and hybrid modeling strategies when machine-learning potentials are deployed in materials discovery settings that cannot avoid out-of-distribution prediction.

The reliability of machine-learned (ML) interatomic potentials in phase space sampling depends critically on whether sampled configurations lie within the interpolative domain of the training data or extend into extrapolative regimes. Despite its central importance, the interpolation–extrapolation distinction remains inconsistently defined across the literature. Existing single-metric approaches—such as convex-hull composition checks, descriptor-space distances, uncertainty estimates, and local-environment similarity—capture only partial aspects of high-dimensional configuration space and frequently yield conflicting classifications. This lack of a rigorous, unified boundary undermines active learning strategies, uncertainty quantification, benchmarking, and the safe deployment of ML potentials in safety-critical applications. This boundary/definitional article introduces a unified and operational framework that delineates the interpolation–extrapolation boundary through four independent and jointly necessary criteria: (C1) compositional coverage, (C2) local-environment similarity, (C3) configurational co-occurrence, and (C4) thermodynamic condition. A configuration is defined as interpolative only when all four criteria are simultaneously satisfied; violation of any criterion constitutes extrapolation. To replace binary classification, the framework further introduces a graded hierarchy of extrapolation—mild, moderate, and severe—based on the number and magnitude of violations. The proposed definitions are deliberately operational, relying exclusively on information available during training-set construction or simulation runtime, and are agnostic to model architecture. Their adoption enables standardized reporting, stratified benchmarking, improved calibration of uncertainty estimates, and more targeted active-learning workflows. By establishing a clear and reproducible boundary, the framework provides a principled foundation for evaluating model reliability during phase space exploration. This work advances the epistemic rigor of ML-driven materials modeling and supports the responsible and transparent deployment of data-driven interatomic potentials.

Out-of-distribution (OOD) generalization remains one of the most pressing barriers to the reliable deployment of artificial intelligence in materials science. Although machine-learning models now routinely achieve sub-0.1 eV/atom errors on in-distribution test sets for formation energies, band gaps, and elastic moduli, these same models frequently collapse when confronted with materials that lie outside the training distribution. Real-world materials discovery and process optimization demand predictions for unseen compositions, novel crystal prototypes, altered thermodynamic conditions, and non-equilibrium dynamical regimes—scenarios that constitute domain shift rather than simple interpolation. This systematic review synthesizes the literature on OOD generalization in materials AI drawing exclusively peer-reviewed publications from high-impact venues including npj Computational Materials, Digital Discovery, Machine Learning: Science and Technology, and Journal of Chemical Theory and Computation. We identify four primary types of domain shift—compositional, structural, thermodynamic, and dynamical—and introduce a fifth multi-dimensional category that captures the realistic superposition of shifts encountered in practice. Current methodological families are critically assessed: domain adaptation (distribution alignment), invariant learning (IRM and Group DRO), physics-informed and symmetry-aware data augmentation, uncertainty quantification for OOD detection, and extrapolation-aware architectures (equivariant networks, multi-fidelity models, and generative priors). Empirical findings across the corpus reveal a consistent pattern: modest gains (20–50 % error reduction) are achievable for small, single-axis shifts, yet performance degrades sharply—and often catastrophically—for large compositional jumps, prototype changes, or combined multi-dimensional shifts. No method currently delivers reliable extrapolation beyond the convex hull of the training manifold. Key gaps persist. The community lacks standardized OOD benchmarks with controlled shift axes, theoretical guarantees for extrapolation remain underdeveloped, and conditional (per-input) robustness guarantees are almost entirely absent. Evaluation metrics are inconsistent, rendering cross-paper comparisons unreliable. This review therefore provides not only a taxonomy and synthesis but also a forward-looking identification of unsolved problems that must be addressed before materials AI can transition from laboratory demonstration to industrial reliability. OOD generalization in materials AI is no longer an optional research direction; it is the central unsolved challenge that will determine the field’s practical impact over the next decade.

Extrapolation in disordered materials such as glasses, amorphous solids, and liquids is fundamentally harder than in crystalline systems. Disordered materials lack periodic symmetry, exhibit highly heterogeneous local environments, and possess variable system sizes that range from hundreds to hundreds of thousands of atoms. These characteristics create an exceptionally large hypothesis space for machine learning models and render pure statistical learning approaches unreliable beyond the training distribution. At the same time, traditional physics-based models remain too approximate for quantitative accuracy in complex disordered systems. This conceptual framework proposes a unified model that bridges statistical learning and physics priors to overcome these limitations and enable reliable extrapolation in disordered materials. The framework rests on three core physics priors—locality, smoothness, and invariance—that act as powerful inductive biases. Locality limits interactions to finite cutoffs, smoothness ensures continuous property landscapes, and invariance (rotational, permutation, and size extensivity) dramatically reduces the effective search space. These priors are embedded directly into statistical learning architectures so that the overall prediction combines a physically grounded baseline with data-driven residual corrections. In extrapolation regimes the physics-informed baseline dominates, producing graceful degradation rather than arbitrary outputs. The model further integrates statistical learning components including uncertainty quantification to flag risky predictions, active learning to expand the training distribution adaptively, multi-fidelity strategies that leverage cheap physics approximations, representation learning for cross-system transfer, and ensemble methods for robustness. The resulting conceptual taxonomy clarifies why extrapolation fails in disordered materials and supplies explicit design principles for extrapolation-aware models. This unified approach shifts the default paradigm in computational materials engineering from purely data-driven or purely physics-driven methods toward a hybrid that respects physical constraints while retaining the flexibility of statistical learning. The framework is expected to accelerate discovery in glass design, amorphous polymers, and liquid electrolytes where out-of-distribution generalization is essential. By treating physics priors as non-optional architectural elements rather than optional regularizers, the model offers a practical path toward trustworthy machine learning predictions in the disordered realm.