Deformable linear objects (DLOs), such as rods, cables, and ropes, play important roles in daily life. However, manipulation of DLOs is challenging as large geometrically nonlinear deformations may occur during the manipulation process. This problem is made even more difficult as the different deformation modes (e.g., stretching, bending, and twisting) may result in elastic instabilities during manipulation. In this paper, we formulate a physics-guided data-driven method to solve a challenging manipulation task -- accurately deploying a DLO (an elastic rod) onto a rigid substrate along various prescribed patterns. Our framework combines machine learning, scaling analysis, and physical simulations to develop a physics-based neural controller for deployment. We explore the complex interplay between the gravitational and elastic energies of the manipulated DLO and obtain a control method for DLO deployment that is robust against friction and material properties. Out of the numerous geometrical and material properties of the rod and substrate, we show that only three non-dimensional parameters are needed to describe the deployment process with physical analysis. Therefore, the essence of the controlling law for the manipulation task can be constructed with a low-dimensional model, drastically increasing the computation speed. The effectiveness of our optimal control scheme is shown through a comprehensive robotic case study comparing against a heuristic control method for deploying rods for a wide variety of patterns. In addition to this, we also showcase the practicality of our control scheme by having a robot accomplish challenging high-level tasks such as mimicking human handwriting, cable placement, and tying knots.

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