Tensor network approaches for learning non-linear dynamical laws
This work addresses the challenge of automated theory building for complex physical systems, offering a physics-informed approach that improves over purely data-driven methods by exploiting fundamental principles.
The authors tackled the problem of identifying non-linear governing equations from physical system observations by developing a tensor network-based method that incorporates physical constraints like locality, enabling scalable and interpretable model learning. They demonstrated that efficient rank-adaptive optimization can learn optimal models without prior knowledge of tensor ranks, balancing expressivity and scalability.
Given observations of a physical system, identifying the underlying non-linear governing equation is a fundamental task, necessary both for gaining understanding and generating deterministic future predictions. Of most practical relevance are automated approaches to theory building that scale efficiently for complex systems with many degrees of freedom. To date, available scalable methods aim at a data-driven interpolation, without exploiting or offering insight into fundamental underlying physical principles, such as locality of interactions. In this work, we show that various physical constraints can be captured via tensor network based parameterizations for the governing equation, which naturally ensures scalability. In addition to providing analytic results motivating the use of such models for realistic physical systems, we demonstrate that efficient rank-adaptive optimization algorithms can be used to learn optimal tensor network models without requiring a~priori knowledge of the exact tensor ranks. As such, we provide a physics-informed approach to recovering structured dynamical laws from data, which adaptively balances the need for expressivity and scalability.