By conceiving physical systems as 3D many-body point clouds, geometric graph
neural networks (GNNs), such as SE(3)/E(3) equivalent GNNs, have showcased
promising performance. In particular, their effective message-passing mechanics
make them adept at modeling molecules and crystalline materials. However,
current geometric GNNs only offer a mean-field approximation of the many-body
system, encapsulated within two-body message passing, thus falling short in
capturing intricate relationships within these geometric graphs. To address
this limitation, tensor networks, widely employed by computational physics to
handle manybody systems using high-order tensors, have been introduced.
Nevertheless, integrating these tensorized networks into the message-passing
framework of GNNs faces scalability and symmetry conservation (e.g.,
permutation and rotation) challenges. In response, we introduce an innovative
equivariant Matrix Product State (MPS)-based message-passing strategy, through
achieving an efficient implementation of the tensor contraction operation. Our
method effectively models complex many-body relationships, suppressing
mean-field approximations, and captures symmetries within geometric graphs.
Importantly, it seamlessly replaces the standard message-passing and
layer-aggregation modules intrinsic to geometric GNNs. We empirically validate
the superior accuracy of our approach on benchmark tasks, including predicting
classical Newton systems and quantum tensor Hamiltonian matrices. To our
knowledge, our approach represents the inaugural utilization of parameterized
geometric tensor networks.

通过创造物理系统的 3D 多体点云，我们提出了一种新型的基于等变矩阵乘积态 (MPS) 的消息传递策略，有效地建模复杂的多体关系并捕捉了几何图中的对称性，超越了现有的几何图神经网络的平均场近似，并在预测经典牛顿系统和量子张量哈密顿矩阵等基准任务上验证了其卓越的准确性，堪称参数化几何张量网络的创新应用。