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Thermal transport, specifically lattice thermal conductivity, in disordered materials such as glasses, high-entropy alloys, and polymers is critical for applications ranging from thermoelectric energy conversion and thermal barrier coatings to electronic cooling and flexible electronics. The structural disorder inherent in these systems breaks translational symmetry, generates heterogeneous local atomic environments, and produces complex, non-perturbative phonon scattering that defies conventional analytical or simulation-based prediction. Equivariant graph neural networks (GNNs) have demonstrated strong empirical promise in this domain by embedding specific inductive biases directly into their architecture. This paper delivers a purely theoretical analysis of the inductive biases that are necessary and sufficient for reliable thermal transport prediction in disordered materials using equivariant GNNs. We formally define inductive bias as any assumption that constrains the hypothesis space and thereby enables generalization from limited data. Through conceptual arguments we identify three key biases—rotational equivariance, locality (finite cutoff), and energy conservation—as fundamental, while showing how supporting biases of smoothness, permutation invariance, and size extensivity reinforce them. Rotational equivariance ensures that tensorial thermal conductivity transforms correctly under arbitrary material rotations; locality reflects the short-range nature of phonon scattering even in globally disordered systems; and energy conservation guarantees that derived forces remain physically consistent for phonon-frequency calculations. We prove conceptually that these biases collectively reduce sample complexity by shrinking the effective dimension of the hypothesis space, improve extrapolation to new disorder realizations provided local environments remain statistically similar, and enforce physical consistency without requiring explicit regularization. Disordered materials constitute a worst-case regime for machine learning because they lack the global symmetries that crystalline systems implicitly exploit; the biases therefore become not merely advantageous but essential. The analysis concludes with direct implications for GNN architecture design, establishing equivariant GNNs as the theoretically preferred framework for thermal transport in disordered materials. This work supplies the missing theoretical foundation that explains why equivariant architectures succeed where generic models fail and offers precise guidance for future model development in computational materials engineering.

Graph-based machine learning has reached maturity for crystalline materials, where periodic boundary conditions and fixed unit cells allow efficient message passing and property prediction. Disordered systems—amorphous solids, liquids, glasses, and polymers—represent the next frontier. These materials dominate everyday technologies, from smartphone screens and optical fibers to electrolytes in batteries and structural alloys, yet they lack the translational symmetry that simplifies graph construction in crystals. This review synthesizes exactly 35 peer-reviewed publications from 2017 to 2025 that apply graph-based learning, graph neural networks (GNNs), and related data-driven methods to non-crystalline materials. The selected works span key target journals including npj Computational Materials, Physical Review B, Journal of Chemical Theory and Computation, Machine Learning: Science and Technology, and Physical Review Materials. The review is organized around a taxonomy of five classes of disordered systems: amorphous solids, liquids, glasses (including supercooled liquids), polymers, and crystals with substitutional disorder. It identifies seven core challenges that distinguish disordered systems from crystals: absence of periodic boundaries, variable graph sizes, heterogeneous local atomic environments, the critical role of medium-range order (5–15 Å), dynamic graph evolution in liquids, lack of standardized benchmarks, and prohibitive computational cost for large simulation boxes. Current methodological approaches are grouped into six categories—local descriptors (non-graph), graph pooling for size invariance, multi-scale GNNs, equivariant architectures, temporal GNNs for liquids, and transfer learning from crystal data—each evaluated for strengths, limitations, and empirical performance on properties such as density, elastic moduli, glass-forming ability, relaxation dynamics, and defect identification. Despite notable progress, three persistent gaps remain: limited size transferability across simulation-box lengths, inadequate capture of medium-range structural correlations, and under-development of methods for dynamic properties. By critically analyzing these 35 studies, this review provides a systematic framework for graph-based learning in disordered materials and highlights open problems that must be solved before these techniques can achieve the same reliability and scalability already demonstrated for crystals. The field stands at an inflection point: continued innovation in graph representations and benchmarking will determine whether graph-based methods can fully unlock predictive modeling for the disordered materials that underpin modern technology.

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.