Density Functional Theory (DFT) is a pivotal method within quantum chemistry and materials science, with its core involving the construction and solution of the Kohn-Sham Hamiltonian. Despite its importance, the application of DFT is frequently limited by the substantial computational resources required to construct the Kohn-Sham Hamiltonian. In response to these limitations, current research has employed deep-learning models to efficiently predict molecular and solid Hamiltonians, with roto-translational symmetries encoded in their neural networks. However, the scalability of prior models may be problematic when applied to large molecules, resulting in non-physical predictions of ground-state properties. In this study, we generate a substantially larger training set (PubChemQH) than used previously and use it to create a scalable model for DFT calculations with physical accuracy. For our model, we introduce a loss function derived from physical principles, which we call Wavefunction Alignment Loss (WALoss). WALoss involves performing a basis change on the predicted Hamiltonian to align it with the observed one; thus, the resulting differences can serve as a surrogate for orbital energy differences, allowing models to make better predictions for molecular orbitals and total energies than previously possible. WALoss also substantially accelerates self-consistent-field (SCF) DFT calculations. Here, we show it achieves a reduction in total energy prediction error by a factor of 1347 and an SCF calculation speed-up by a factor of 18\%. These substantial improvements set new benchmarks for achieving accurate and applicable predictions in larger molecular systems.

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.

Graph classification, which aims to predict a label for a graph, has many real-world applications such as malware detection, fraud detection, and healthcare. However, many studies show an attacker could carefully perturb the structure and/or node features in a graph such that a graph classifier misclassifies the perturbed graph. Such vulnerability impedes the deployment of graph classification in security/safety-critical applications. Existing empirical defenses lack formal robustness guarantees and could be broken by adaptive or unknown attacks. Existing provable defenses have the following limitations: 1) they achieve sub-optimal robustness guarantees for graph structure perturbation, 2) they cannot provide robustness guarantees for arbitrarily node feature perturbations, 3) their robustness guarantees are probabilistic, meaning they could be incorrect with a non-zero probability, and 4) they incur large computation costs. We aim to address those limitations in this work. We propose GNNCert, a certified defense against both graph structure and node feature perturbations for graph classification. Our GNNCert provably predicts the same label for a graph when the number of perturbed edges and the number of nodes with perturbed features are bounded. Our results on 8 benchmark datasets show that GNNCert outperforms three state-of-the-art methods.