Physics-informed neural networks with hard constraints for inverse design

TL;DR

This work presents hPINN, a physics-informed neural network framework that enforces hard PDE and inequality constraints for PDE-constrained inverse design. By embedding Dirichlet and periodic boundary conditions directly in the network architecture and employing penalty and augmented Lagrangian constraint enforcement, hPINN achieves objective parity with adjoint-based PDE solvers while often producing smoother, simpler designs in non-unique design spaces. The holography and Stokes-flow topology-optimization examples demonstrate that hard-constraint formulations can match traditional solver performance with comparable compute cost, and the augmented Lagrangian approach offers robust convergence and constraint satisfaction. Overall, hPINN provides a solver-free, mesh-free alternative for PDE-constrained inverse design with potential for broader application and simplified implementation.

Abstract

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.

Figures (10)

• Figure 1: Physics-informed neural networks with hard-constraint Dirichlet and periodic boundary conditions. (A) Two independent neural networks $\hat{\mathbf{u}}(\mathbf{x};\bm{\theta}_u)$ and $\hat{\gamma}(\mathbf{x};\bm{\theta}_\gamma)$ are conostructed to approximate $\mathbf{u}(\mathbf{x})$ and $\gamma(\mathbf{x})$. The gradients in the PDE-informed loss is computed via AD. (B) Dirichlet BCs are strictly imposed into the network architecture by modifying the network output. (C) Periodic BCs are strictly imposed into the network architecture by modifying the network input.
• Figure 1: Holography problem setup and the neural network architecture. (A) The whole computational domain includes the main domain $\Omega = [-2,-2] \times [2,3]$ and a PML of depth one in the shaded region. The design region for the permittivity $\varepsilon$ is in blue, and the center of the current $J$ is the dashed red line in the domain $\Omega_1$. The target electrical field is defined in $\Omega_3$. (B) The time-harmonic current $J$. (C) The architecture of the hPINN with the Dirichlet and periodic BCs embedded directly in the network. The network inputs are $x$ and $y$, and the outputs are $\Re[E]$, $\Im[E]$ and $\varepsilon$.
• Figure 2: hPINN for solving a forward problem. (A and B) The (A) real part $\Re[E]$ and (B) imaginary part $\Im[E]$ of the hPINN solution. (C and D) The (C) real part $\Re[E]$ and (D) imaginary part $\Im[E]$ of the FDFD solution. (E and F) The (E) real part $\Re[\mathcal{F}[\mathbf{x}]]$ and (F) imaginary part $\Im[\mathcal{F}[\mathbf{x}]]$ of the PDE-informed loss in Eq. \ref{['eq:holography_loss_pde']}. (G and H) The (G) training loss $\mathcal{L}_\mathcal{F}$ and (H) $L^2$ relative error versus the number of optimization iterations. (I) The correlation between the training loss and the $L^2$ relative error during the training process.
• Figure 3: hPINN for the inverse design of holography via the approach of soft constraints. (A) Three examples of random initialization of $\varepsilon$. (B) The objective $\mathcal{J}$ of different designs obtained from hPINNs with different $\mu_\mathcal{F}$. (C, D, and E) The (left top) permittivity $\varepsilon$, (left bottom) training trajectory, (center) hPINN solution $|E|^2$, and (right) the reference $|E|^2$ obtained from FDFD for (C) $\mu_\mathcal{F}=0.1$, (D) $\mu_\mathcal{F}=10$, and (E) $\mu_\mathcal{F}=2$.
• Figure 4: hPINN for the inverse design of holography via the penalty method. (A to D) The results of $\mu_\mathcal{F}^0=2$ and $\beta_\mathcal{F}=2$. (A) The losses versus the number of optimization iterations, (B) the objective value after the network is trained with $\mu_\mathcal{F}^k$, (C) the optimized permittivity function $\varepsilon$ at $k=1$, and (D) the corresponding electric field $|E|^2$. (E) The objective value versus $\mu_\mathcal{F}^k$ when using different values of $\beta_\mathcal{F}$. (F) The total number of optimization iterations (which is proportional to the computational cost) for different $\beta_\mathcal{F}$. (G) The objective value versus $\mu_\mathcal{F}^k$ for different values of $\mu_\mathcal{F}^0$..