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\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . Inverse design arises in a variety of areas in engineering such as acoustic, mechanics, thermal/electronic transport, electromagnetism, and optics. Topology optimization is an important form of inverse design, where one optimizes a designed geometry to achieve targeted properties parameterized by the materials at every point in a design region. This optimization is challenging, because it has a very high dimensionality and is usually constrained by partial differential equations (PDEs) and additional inequalities. Here, we propose a new deep learning method---physics-informed neural networks with hard constraints (hPINNs)---for solving topology optimization. hPINN leverages the recent development of PINNs for solving PDEs, and thus does not require a large dataset (generated by numerical PDE solvers) for training. However, all the constraints in PINNs are soft constraints, and hence we impose hard constraints by using the penalty method and the augmented Lagrangian method. We demonstrate the effectiveness of hPINN for a holography problem in optics and a fluid problem of Stokes flow. We achieve the same objective as conventional PDE-constrained optimization methods based on adjoint methods and numerical PDE solvers, but find that the design obtained from hPINN is often smoother for problems whose solution is not unique. Moreover, the implementation of inverse design with hPINN can be easier than that of conventional methods because it exploits the extensive deep-learning software infrastructure.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . inverse design, topology optimization, partial differential equations, physicsinformed neural networks, penalty method, augmented Lagrangian method \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 35R30, 65K10, 68T20 \bfD \bfO \bfI .

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Two successive decoalescence events in the hydride region of the 1H NMR spectrum of [ReH5(PPh3)2(py)] (py = pyridine) are now firmly associated with turnstile and pseudorotation fluxionality mechanisms by eliminating an alternative pairwise mechanism. Ab intio (B3LYP) calculations on ReH5(PH3)2L (L = pyridine) have located the transition state for the turnstile mechanism, which proves to be a second dodecahedral tautomer of the starting complex with the pyridine in the normally unfavorable A site. The fluxional process can therefore be considered as an interconversion of two dodecahedral tautomers, and the barrier for the process is identical with the energy difference of the two tautomers. From a comparison in ReH5(PPh3)2L (L = 2-(acetylamino)pyridine and 4-(acetylamino)pyridine), it is clear that having a potentially hydrogen-bonding NH group at the ortho or para positions of the pyridine ring causes an acceleration of the fluxionality, as a result of intramolecular Re−H···H−N hydrogen bonding. The theoretical calculations on ReH5(PH3)2L (L = 2-aminopyridine and 4-aminopyridine) show that the experimental barriers are the result of a compromise between two factors:  hydrogen bonding, which lowers the barrier for the 2-amino compound, and H···H repulsion resulting from an excessively close approach of the two H atoms in the transition state, which raises the barrier. This implies that the particular hydrogen-bonding ligands chosen were too rigid for optimal rate acceleration.

Directional light scattering is important in basic research and real applications. This area has been successfully downscaled to wavelength and subwavelength scales with the development of optical antennas, especially single-element nanoantennas. Here, by adding an auxiliary resonant structure to a single-element plasmonic nanoantenna, we show that the highly efficient lowest-order antenna mode can be effectively transferred into inactive higher-order modes. On the basis of this mode conversion, scattered optical fields can be well manipulated by utilizing the interference between different antenna modes. Both broadband directional excitation of surface plasmon polaritons (SPPs) and inversion of SPP launching direction at different wavelengths are experimentally demonstrated as typical examples. The proposed strategy based on mode conversion and mode interference provides new opportunities for the design of nanoscale optical devices, especially directional nanoantennas.