Spin-Weighted Spherical Harmonics Enable Complete and Scalable $\mathrm{E}(3)$-Equivariant Networks

$\mathrm{E}(3)$-equivariant networks are promising for 3D atomistic system modeling, yet their scalability is limited by the $O(L^6)$ complexity of the Clebsch-Gordan Tensor Product (CGTP). The recently proposed Gaunt Tensor Product (GTP) reduces the complexity but is unable to capture the antisymmetric paths, resulting in incomplete expressivity. In this work, we present SpinGTP, an approach to overcome the GTP incompleteness by generalizing from scalar functions to Spin-Weighted Spherical Harmonics (SWSH). By relying on the algebraic properties of SWSH, SpinGTP recovers the missing antisymmetric interactions while maintaining the asymptotic efficiency of GTP. It also allows for a more expressive equivariant basis that naturally accounts for the parity-odd components of tensor products. We evaluate SpinGTP across diverse benchmarks, including Tetris, 3BPA, SPICE-MACE-OFF, and OC20. Our results show that SpinGTP achieves accuracies comparable to full CGTP. Notably, by explicitly capturing antisymmetric paths, SpinGTP exhibits superior performance in tasks involving chiral materials and non-centrosymmetric geometries. This work provides a complete, scalable, and mathematically rigorous path toward high-order equivariance in large-scale 3D atomistic system simulations.

Generative Quasi-Continuum Modeling of Confined Fluids at the Nanoscale

We present a data-efficient, multiscale framework for predicting the density profiles of confined fluids at the nanoscale. While accurate density estimates require prohibitively long timescales that are inaccessible by ab initio molecular dynamics (AIMD) simulations, machine-learned molecular dynamics (MLMD) offers a scalable alternative, enabling the generation of force predictions at ab initio accuracy with reduced computational cost. However, despite their efficiency, MLMD simulations remain constrained by femtosecond timesteps, which limit their practicality for computing long-time averages needed for accurate density estimation. To address this, we propose a conditional denoising diffusion probabilistic model (DDPM) based quasi-continuum approach that predicts the long-time behavior of force profiles along the confinement direction, conditioned on noisy forces extracted from a limited AIMD dataset. The predicted smooth forces are then linked to continuum theory via the Nernst-Planck equation to reveal the underlying density behavior. We test the framework on water confined between two graphene nanoscale slits and demonstrate that density profiles for channel widths outside of the training domain can be recovered with ab initio accuracy. Compared to AIMD and MLMD simulations, our method achieves orders-of-magnitude speed-up in runtime and requires significantly less training data than prior works.