# Efficient and Scalable Density Functional Theory Hamiltonian Prediction through Adaptive Sparsity

arXiv: https://arxiv.org/abs/2502.01171 · HTML: https://mlsanity.com/paper/2502.01171/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 50 references that are in the map.
- First posted on arXiv 3 Feb 2025.

## Builds on (most cited first, 24 of 24)

- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Learning both Weights and Connections for Efficient Neural Networks](https://mlsanity.com/paper/1506.02626/) (2015)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [ChemRL-GEM: Geometry Enhanced Molecular Representation Learning for Property Prediction](https://mlsanity.com/paper/2106.06130/) (2021)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Geometric Deep Learning on Molecular Representations](https://mlsanity.com/paper/2107.12375/) (2021)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [Unifying machine learning and quantum chemistry -- a deep neural network for molecular wavefunctions](https://mlsanity.com/paper/1906.10033/) (2019)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [GemNet-OC: Developing Graph Neural Networks for Large and Diverse Molecular Simulation Datasets](https://mlsanity.com/paper/2204.02782/) (2022)
- [Efficient and Equivariant Graph Networks for Predicting Quantum Hamiltonian](https://mlsanity.com/paper/2306.04922/) (2023)
- [QH9: A Quantum Hamiltonian Prediction Benchmark for QM9 Molecules](https://mlsanity.com/paper/2306.09549/) (2023)
- [Comprehensive Graph Gradual Pruning for Sparse Training in Graph Neural Networks](https://mlsanity.com/paper/2207.08629/) (2022)
- [Self-Consistency Training for Density-Functional-Theory Hamiltonian Prediction](https://mlsanity.com/paper/2403.09560/) (2024)
- [E2Former: An Efficient and Equivariant Transformer with Linear-Scaling Tensor Products](https://mlsanity.com/paper/2501.19216/) (2025)
- [Infusing Self-Consistency into Density Functional Theory Hamiltonian Prediction via Deep Equilibrium Models](https://mlsanity.com/paper/2406.03794/) (2024)
- [GeoMFormer: A General Architecture for Geometric Molecular Representation Learning](https://mlsanity.com/paper/2406.16853/) (2024)

## Built on by (most cited first, 11 of 11)

- [Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systems](https://mlsanity.com/paper/2502.19227/) (2025)
- [High-order Equivariant Flow Matching for Density Functional Theory Hamiltonian Prediction](https://mlsanity.com/paper/2505.18817/) (2025)
- [Learning from the electronic structure of molecules across the periodic table](https://mlsanity.com/paper/2510.00224/) (2025)
- [Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames](https://mlsanity.com/paper/2506.09398/) (2025)
- [Towards A Transferable Acceleration Method for Density Functional Theory](https://mlsanity.com/paper/2509.25724/) (2025)
- [Advancing Universal Deep Learning for Electronic-Structure Hamiltonian Prediction of Materials](https://mlsanity.com/paper/2509.19877/) (2025)
- [M=oLe-Lambda: Learning the Coupled-Cluster Response State for Energies, Gradients, and Properties](https://mlsanity.com/paper/2605.29622/) (2026)
- [Coupled Cluster con M=oLe: Molecular Orbital Learning for Neural Wavefunctions](https://mlsanity.com/paper/2602.20232/) (2026)
- [A Fixed-Point Neural Operator for Size- and Functional-Transferable Hamiltonian Prediction](https://mlsanity.com/paper/2606.14498/) (2026)
- [Equivariant Flow Matching for Electron Density Prediction](https://mlsanity.com/paper/2610.02651/) (2026)
- [Physics-Aligned Electronic Ground-State Learning Improves Generalization](https://mlsanity.com/paper/2610.10298/) (2026)

## Abstract (arXiv)

Hamiltonian matrix prediction is pivotal in computational chemistry, serving as the foundation for determining a wide range of molecular properties. While SE(3) equivariant graph neural networks have achieved remarkable success in this domain, their substantial computational cost--driven by high-order tensor product (TP) operations--restricts their scalability to large molecular systems with extensive basis sets. To address this challenge, we introduce SPHNet, an efficient and scalable equivariant network, that incorporates adaptive SParsity into Hamiltonian prediction. SPHNet employs two innovative sparse gates to selectively constrain non-critical interaction combinations, significantly reducing tensor product computations while maintaining accuracy. To optimize the sparse representation, we develop a Three-phase Sparsity Scheduler, ensuring stable convergence and achieving high performance at sparsity rates of up to 70%. Extensive evaluations on QH9 and PubchemQH datasets demonstrate that SPHNet achieves state-of-the-art accuracy while providing up to a 7x speedup over existing models. Beyond Hamiltonian prediction, the proposed sparsification techniques also hold significant potential for improving the efficiency and scalability of other SE(3) equivariant networks, further broadening their applicability and impact. Our code can be found at https://github.com/microsoft/SPHNet.