  Inverse design arises in a variety of areas in engineering such as acoustic,
mechanics, thermal/electronic transport, electromagnetism, and optics. Topology
optimization is a major form of inverse design, where we optimize a designed
geometry to achieve targeted properties and the geometry is parameterized by a
density function. 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 rely on any numerical PDE solver. 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 simpler and
smoother for problems whose solution is not unique. Moreover, the
implementation of inverse design with hPINN can be easier than that of
conventional methods.
