Differentiable Programming Tensor Networks

Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and trains them using automatic differentiation (AD). The concept emerges from deep learning but is not only limited to training neural networks. We present theory and practice of programming tensor network algorithms in a fully differentiable way. By formulating the tensor network algorithm as a computation graph, one can compute higher order derivatives of the program accurately and efficiently using AD. We present essential techniques to differentiate through the tensor networks contractions, including stable AD for tensor decomposition and efficient backpropagation through fixed point iterations. As a demonstration, we compute the specific heat of the Ising model directly by taking the second order derivative of the free energy obtained in the tensor renormalization group calculation. Next, we perform gradient based variational optimization of infinite projected entangled pair states for quantum antiferromagnetic Heisenberg model and obtain start-of-the-art variational energy and magnetization with moderate efforts. Differentiable programming removes laborious human efforts in deriving and implementing analytical gradients for tensor network programs, which opens the door to more innovations in tensor network algorithms and applications.Comment: Typos corrected, discussion and refs added; revised version accepted for publication in PRX. Source code available at https://github.com/wangleiphy/tensorgra

Composite Rotor Blade Design Optimization for Vibration Reduction with Aeroelastic Constraints

AbstractThe paper presents an analytical study of the helicopter rotor vibratory load reduction design optimization with aeroelastic stability constraints. The composite rotor blade is modeled by beam type finite elements, and warping deformation is taken into consideration for 2 dimension analysis, while the one-dimension nonlinear differential equations of blade motion are formulated via Hamilton's principle. The rotor hub vibratory loads is chosen as the objective function, while rotor blade section construction parameter, composite material ply structure and blade tip swept angle as the design variables, and autorotation inertia, natural frequency and aeroelastic stability as the constraints. A 3 bladed rotor is designed, as an example, based on the vibratory hub load reduction optimization process with swept tip angle and composite material. The calculating results show a 24. 9%-33% reduction of 3/rev hub loads in comparison with the base-line rotor

Conserved quantities for asymptotically AdS spacetimes in quadratic curvature gravity in terms of a rank-4 tensor

We investigate the conserved quantities associated to Killing isometries for asymptotically AdS spacetimes within the framework of quadratic-curvature gravity. By constructing a rank-4 tensor possessing the same index symmetries as the ones of the Riemann tensor, we propose a 2-form potential resembling the Noether one for quadratic-curvature gravity. Such a potential is compared with the results via other methods existing in the literature to establish the equivalence. Then this potential is adopted to define conserved quantities of asymptotically AdS spacetimes. As applications, we explicitly compute the mass of static spherically-symmetric spacetimes, as well as the mass and the angular momentum for rotating spacetimes, such as the four(higher)-dimensional Kerr-AdS black holes and black strings embedded in quadratic-curvature gravities. Particularly, we emphasize the conserved charges of Einstein-Gauss-Bonnet, Weyl and critical gravities, together with the ones for the asymptotically AdS solutions satisfying vacuum Einstein field equations.Comment: 50 pages, no figures, accepted by PR