Tensor Network: from the Perspective of AI4Science and Science4AI

Tensor network has been a promising numerical tool for computational problems across science and AI. For their emerging and fast development especially in the intersection between AI and science, this paper tries to present a compact review, regarding both their applications and its own recent technical development including open-source tools. Specifically, we make the observations that tensor network plays a functional role in matrix compression and representation, information fusion, as well as quantum-inspired algorithms, which can be generally regarded as Science4AI in our survey. On the other hand, there is an emerging line of research in tensor network in AI4Science especially like learning quantum many-body physics by using e.g. neural network quantum state. Importantly, we unify tensorization methodologies across classical and modern architectures, and particularly show how tensorization bridges low-order parameter spaces to high-dimensional representations without exponential parameter growth, and further point out their potential use in scientific computing. We conclude the paper with outlook for future trends.