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.
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