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Page 12 of 23 Li et al. J. Mater. Inf. 2025, 5, 43 https://dx.doi.org/10.20517/jmi.2025.17

Beyond architectural innovations, accuracy improvements have also come from better loss function design.
WANet introduces the physically-motivated Wavefunction Alignment Loss (WALoss), which surpasses
simple element-wise error metrics by directly enforcing alignment between predicted and ground-truth
[75]
wavefunctions. Additionally, TraceGrad , a lightweight plug-in module, enhances accuracy using SO(3)-
invariant trace quantities as supervisory signals which have been integrated with QHNet and DeepH-E3
[76]
[67]
[Figure 6A]. This approach guides the development of high-quality equivariant representations, achieving
good prediction performance.

Looking ahead, the evolution of DHNNs will hinge on three interlinked challenges. First, achieving true
universality and scalability demands that DHNNs generalize across the full periodic table and extend to very
larger systems. This is a goal that will require innovations in strictly local message passing, hierarchical
graph decompositions and delta-learning schemes to drive computational cost toward linear scaling.
Second, broadening the applicability of these architectures to encompass richer physical phenomena from
spin-orbit coupling and non-collinear magnetism to excited state dynamics and quantum transport
[Figure 4B]. This will call for modular and multi-task frameworks capable of integrating additional
Hamiltonian terms and perturbative operators without sacrificing symmetry rigor. Finally, unlocking the
full scientific potential of DHNNs rests on enhancing interpretability and trust, which is essential for
mechanistic insight, model validation and widespread adoption. It needs embedding physics-informed
constraints, developing graph-based explainability tools and rigorous uncertainty quantification [Table 2].

Deep Hamiltonian model: generality and scalability
DHNNs have rapidly emerged as a transformative approach for surmounting the steep computational
demands of conventional quantum-mechanical methods such as Kohn-Sham DFT [Table 2]. Yet despite
their promising accuracy and efficiency, DHNNs remain in an infant stage. Their evolution into a truly
universal tool for materials science and quantum chemistry depends on overcoming two intertwined
challenges. One is the generality that the capacity to learn Hamiltonians across the full breadth of chemical
space and structural complexity. The other is the scalability, which is the ability to maintain accuracy and
computational efficiency as system size grows. Addressing these twin imperatives is essential to unlock
DHNN potential as a high-throughput, first-principles surrogate for large-scale quantum simulations.

Despite significant progress by architectures such as DeepH and HamGNN toward broader material
coverage, existing DHNNs still struggle with true generality [Figure 6A and B]. To date, these models have
been developed and validated almost exclusively on well-ordered crystalline systems, leaving out inherently
[88]
[89]
[90]
disordered amorphous networks , defected materials and high-entropy alloys , that pervade real-world
materials. Disordered systems, for example, are ubiquitous in technologically relevant materials. However,
these systems lack the long-range translational symmetry characteristic of well-ordered crystals, and they
typically exhibit high configurational entropy. This combination results in a vast and highly diverse
landscape of local atomic environments, where each atomic neighborhood can be distinct. The current
message-passing schemes predominantly employed by DHNNs inherently operate with limited receptive
fields. These schemes are designed to learn relationships by primarily considering local atomic structures
when mapping to Hamiltonian matrix elements. Consequently, such models struggle to effectively capture
the nuances of structural disorder that extend beyond these local regions. Such structural disorder poses a
formidable challenge: without mechanisms to capture the irregular local environments and stochastic
atomic arrangements characteristic of these systems, DHNNs risk losing both robustness and transferability.
Closing this gap by integrating disorder aware descriptors, adaptive message passing schemes and targeted
data augmentation will be essential for DHNNs to become universally reliable electronic structure
surrogates.