Equivariant Atomic and Lattice Modeling Using Geometric Deep Learning for Crystal Structure Optimization

Structure optimization, which yields the relaxed structure (minimum-energy state), is essential for reliable materials property calculations, yet traditional ab initio approaches such as density-functional theory (DFT) are computationally intensive. Machine learning (ML) has emerged to alleviate this bottleneck but suffers from two major limitations: (i) existing models operate mainly on atoms, leaving lattice vectors implicit despite their critical role in structural optimization; and (ii) they often rely on multi-stage, non-end-toend workflows that are prone to error accumulation. Here, we present E 3 Relax-an end-to-end equivariant graph neural network that maps an unrelaxed crystal directly to its relaxed structure. E 3 Relax promotes both atoms and lattice vectors to graph nodes endowed with dual scalar-vector features, enabling unified and symmetry-preserving modeling of atomic displacements and lattice deformations. A layer-wise supervision strategy forces every network depth to make a physically meaningful refinement, mimicking the incremental convergence of DFT while preserving a fully end-to-end pipeline. We evaluate E 3 Relax on four benchmark datasets and demonstrate that it achieves remarkable accuracy and efficiency. Through DFT validations, we show that the structures predicted by E 3 Relax are energetically favorable, making them suitable as high-quality initial configurations to accelerate DFT calculations. Our code and data are available at https://github.com/Shen-Group/E3Relax .