Mandala is a modular software framework for learning block-sparse electronic-structure matrices with E(3)-equivariant graph neural networks that 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.
Abstract
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.
This work proposes an operator-centric framework in which the external (nuclear) potential, expressed in an AO basis, serves as the model input and builds hierarchical, body-ordered representations of atomic configurations that closely mirror the principles underlying several popular atom-centered descriptors.
Jigyasa Nigam, T. Smidt, G. Dusson· Journal of Chemical Physics· 2 citations
Equivariant graph neural networks have proven effective tools for inference of material's properties directly from their structure. Traditionally, these have been applied such that they respect full $O(3)$ equivariance, so that any rotation or reflection of the input structure is respected in the model's output. While this works for general arrangements of atoms, additional symmetries of atomistic systems are left unleveraged. Furthermore, any symmetries of the filter functions are implicitly learned from the full dataset and not strictly enforced. In this work, we introduce point-group symmetry aware equivariant graph neural networks (PGEqNN) for materials science, with filter functions aligned with symmetry-aware indices for greater granularity in predictive tasks. With this architecture, we show that most of the predictive power of equivariant networks for tensorial elastic and dielectric datasets lies in the trivial subspaces of the point-group adapted bases. Exploiting this, an $A_1$-restricted variant matches or improves on its full point-group and $SO(3)$-partitioned counterparts while training fewer active parameters, yielding leaner models of equal accuracy.
Accurate ab initio molecular dynamics (AIMD) simulations of complex, fluxional chemical systems are severely limited by the high computational scaling of correlated electronic structure methods. To overcome this bottleneck, we present a robust, graph-theoretic molecular fragmentation framework integrated with machine learning to directly model post-Hartree-Fock nuclear forces at coupled cluster accuracy. Bypassing the limitations of automatic differentiation on learned energy surfaces that may struggle with link-atom Jacobians, our approach directly predicts nuclear force vectors. By projecting these vectors onto fragment-fixed principal axes of inertia, we establish co-variant descriptors that naturally preserve rotational, translational, and permutational invariance. The methodology achieves exceptional high parameter efficiency through a vector-valued training protocol that reduces trainable parameters by over an order of magnitude, while an unsupervised mini-batch k-means space tessellation algorithm constructs highly representative training databases using only 10% to 20% of reference configurations. We rigorously validated this framework on the highly fluxional solvated Zundel cation H_{13}O_6^+ ). Our fully machine-learning-predicted AIMD trajectories successfully reproduced complex dynamical signatures and key structural characteristics, including radial distribution functions and the velocity autocorrelation power spectrum. Ultimately, this scalable, systematically improvable framework bridges the gap between high-level correlated wavefunction theories and long-timescale reactive sampling, laying the foundation for advanced, LLM-inspired transfer learning in modern chemical dynamics simulations.
Elastic Network Models (ENMs), particularly the Gaussian Network Model (GNM) and its distance-weighted variant (mENM), predict per-residue protein flexibility from Cα contact graphs at low computational cost. Their central limitation is the assumption of uniform spring constants, which ignores the chemical identity, burial depth, and evolutionary conservation of individual residue contacts. We introduce the Learnable Graph Network Model (LGNM), a heterogeneous ENM in which per-edge spring constants θij = fi · fj · (dc/rij)2 are parameterised by per-residue flexibility coefficients {fi} predicted by a physics-constrained Graph Neural Network (GNN). The GNN is trained on molecular dynamics (MD)-derived root-mean-square fluctuation (RMSF) profiles from 413 proteins in the ATLAS database, using fold-disjoint CATH superfamily splits. The learning objective is an instance of the Quantum Neural PDE (QNPDE) Hamiltonian learning framework, with K = 3 operator types enabling an O(K) quantum gradient versus O (N3) classical pseudo-inversion. On 91 held-out test proteins, LGNM achieves mean per-protein Pearson correlation r = 0.8549± 0.1055, versus r = 0.8024 ± 0.1167 for mENM (Δr = +0.0525; 77/91 proteins improved). The implementation of this methedology is aviliable at https://lgnm.compbiosysnbu.in/ allowing researchers to evaluate flexibility and downstream processses.
A novel anisotropic machine learning CG potential is introduced that extends the point particle representation of atomic nuclei to massive ellipsoidal beads with orientation-dependent features, enabling the learning of energies, forces, and torques directly from atomistic data.
Accurate prediction of electronic Hamiltonians would enable broad property inference while avoiding the high computational cost of Density Functional Theory (DFT). However, progress toward general-purpose materials foundation models is limited by a data bottleneck: existing Hamiltonian datasets are typically small, lack structural diversity, and often omit essential relativistic physics such as spin--orbit coupling (SOC). We therefore construct UniHam, a large-scale Hamiltonian dataset and benchmark suite comprising 100,000+ DFT-computed complex-valued Hermitian Hamiltonians with full SOC, covering 72 elements and a wide range of crystal geometries and symmetries (spanning diverse lattice types and space-group families). Building on UniHam, we benchmark two representative state-of-the-art models under a standardized protocol and introduce complementary evaluation metrics that jointly assess three dimensions: (i) Hamiltonian reconstruction accuracy, (ii) out-of-distribution (OOD) generalization across composition/symmetry shifts, and (iii) the ability to support downstream property prediction from the predicted Hamiltonians. Experiments on UniHam demonstrate that the proposed benchmark and metrics effectively differentiate model capabilities, revealing intrinsic SOC- and element-dependent failure modes, large variations in compositional OOD robustness, and the necessity of spectral-level evaluation to assess whether Hamiltonian predictions reliably support downstream electronic-structure properties. Overall, UniHam provides a reproducible, SOC-complete benchmark that can sharpen model comparisons and accelerate the development of next-generation foundation models for quantum materials.
Yuewen Huang, Pin Chen, Yutong Lu· Proceedings of the 32nd ACM...· 0 citations