Physics-Informed Neural Networks with Hard Constraints for Inverse Design

\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