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This paper introduces Mandala, a modular software framework that leverages E(3)-equivariant graph neural networks to learn block-sparse electronic-structure matrices, addressing the limitations of traditional machine-learning interatomic potentials that only predict energies and forces. By providing a unified representation of quantum-mechanical operators, Mandala enables the direct evaluation of critical electronic observables such as band energy and density of states, thereby enhancing the modeling of complex materials. The framework's design allows for flexibility in handling diverse chemical compositions and neural architectures, making it a valuable tool for large-scale electronic-structure calculations in materials science and chemistry.
Mandala connects electronic-structure learning to observable-guided modeling, enabling direct predictions of critical quantum observables that traditional MLIPs overlook.
Electronic-structure calculations based on Kohn-Sham density functional theory remain indispensable in computational materials science and chemistry. Their computational cost, however, limits accessible system sizes and simulation times. At the same time, conventional machine-learning interatomic potentials (MLIPs), which are becoming the workhorse of large-scale materials modeling, usually target only energies and forces. They therefore leave out the quantum-operator-level information required to reconstruct band structures, densities of states, spatial charge distributions, and other electronic observables. \texttt{Mandala} fills this methodological gap. It is a modular software framework for learning block-sparse electronic-structure matrices with E(3)-equivariant graph neural networks. The framework is built around a unified representation of atom-resolved Hamiltonian, overlap, and density matrices, together with reusable abstractions for basis conversion, sparse block handling, irreducible representation mapping, graph construction, model definition, and training. This design allows \texttt{Mandala} to support heterogeneous chemical compositions, a wide range of neural architecture variants within one workflow, and multiple electronic-structure backends. \texttt{Mandala} evaluates selected observables directly from the predicted operators, including band energy, electron count, density of states, and band structure. This connects electronic-structure learning and observable-guided modeling while retaining a representation tied to quantum-mechanical operators rather than only scalar or vector targets as in MLIPs. In this form, \texttt{Mandala} is intended to complement atomistic interatomic potential workflows by resolving electronic structure and operator-derived observables within one scalable implementation.
