Tensor Decomposition Networks for Accelerating Machine Learning Force Field Computations
Yuchao Lin, Cong Fu, Zachary Krueger, Haiyang Yu, Maho Nakata, Jianwen Xie, Emine Kucukbenli, Xiaofeng Qian, Shuiwang Ji
$\rm{SO}(3)$-equivariant networks are the dominant models for machine learning force fields (MLFF). The key operation of such networks is the Clebsch-Gordan (CG) tensor product, which is computationally expensive. To accelerate the computation, we develop tensor decomposition networks (TDNs) as a class of approximately equivariant networks whose CG tensor products are replaced by low-rank tensor decompositions, such as the CANDECOMP/PARAFAC (CP) decomposition. With the CP decomposition, we prove (i) a uniform bound on the induced error of $\rm{SO}(3)$-equivariance, and (ii) the universality of approximating any equivariant bilinear map. To further reduce the number of parameters, we propose path-weight sharing that ties all multiplicity-space weights across the $\mathcal{O}(L^3)$ CG paths into a single path without compromising equivariance, where $L$ is the maximum angular degree. The resulting layer acts as a plug-and-play replacement for tensor products in existing networks, and the computational complexity of tensor products is reduced from $\mathcal{O}(L^6)$ to $\mathcal{O}(L^4)$. We evaluate TDNs on a newly curated, DFT-calculated molecular force field relaxation dataset with 105 million snapshots. We also use existing datasets, including OC20, and OC22. Results show that TDNs achieve competitive performance with dramatic speedup in computations.