Source-linked AI summary
Physics-informed neural networks with hard constraints for inverse design
Lu Lu, Raphael Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, Steven G. Johnson
TL;DR
High-dimensional PDE-constrained inverse design must optimize geometry while satisfying PDEs and inequality constraints. The paper proposes hPINN, combining PINNs with hard-constraint penalty and augmented Lagrangian methods. Across holography and Stokes-flow examples, hPINN matches conventional objective performance and often yields simpler, smoother designs.
Problem
PDE-constrained topology optimization involves very high-dimensional geometries subject to PDEs and additional inequalities.
Method
hPINN uses physics-informed neural networks with penalty and augmented Lagrangian methods to impose inverse-design constraints without numerical PDE solvers.
Results
hPINN achieves the same objective-function performance as conventional adjoint-based PDE-constrained optimization while often producing simpler and smoother designs.
Takeaways & Limitations
hPINN provides a purely deep-learning, one-size-fit-all implementation for inverse design.
Takeaways & Limitations
Soft constraints may violate PDE or inequality constraints, while large penalty coefficients can cause ill-conditioned optimization or prevent progress on the design objective.
Abstract
from arXiv · showhide
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.
1. Introduction.
PDE-constrained inverse design must optimize high-dimensional geometries while satisfying PDEs and additional inequalities. The paper introduces hPINN, which imposes these constraints for inverse design using neural networks.
- PDE-constrained topology optimization is important across computational science and engineering, including optics, fluids, and solid mechanics.
- Traditional approaches use numerical PDE solvers and adjoint methods to optimize large numbers of design parameters.
- PINNs offer a meshfree alternative for solving PDEs and have been applied to diverse inverse problems.
- High-dimensional design spaces and manufacturing-related inequality constraints make inverse design challenging.
- hPINN imposes equality and inequality constraints through penalty and augmented Lagrangian methods, with soft constraints included for comparison.
- In holography and Stokes flow, hPINN achieves the same objective-function performance as conventional methods while often producing simpler, smoother designs.
2. Methods.
The method represents both PDE solutions and design parameters with neural networks, enforces boundary conditions architecturally, and handles remaining constraints through soft, penalty, or augmented Lagrangian formulations.
- 2.1. Inverse design.: The inverse-design formulation minimizes an objective J over the design parameter γ while enforcing PDE, boundary, and inequality constraints.
- 2.2. Physics-informed neural networks.: The PDE solution u and design parameter γ are approximated by separate fully connected neural networks trained with PDE residual losses.
- 2.2. Physics-informed neural networks.: Automatic differentiation evaluates network-output derivatives at residual points without generating a mesh.
- 2.3. Hard-constraint boundary conditions.: Dirichlet boundary conditions are enforced by modifying network outputs, while periodic conditions are enforced by replacing inputs with Fourier basis functions.
- 2.4. Soft constraints.: Soft constraints penalize PDE and inequality violations, but penalty weights create a trade-off between feasibility and ill-conditioning.
- 2.5. Penalty method.: The penalty method can converge slowly or become trapped at poor local minima when its hyperparameters are poorly scaled.
- 2.6. Augmented Lagrangian method.: The augmented Lagrangian method avoids sending penalty coefficients to infinity and has a considerably better convergence rate than the penalty method.
3. Results.
The results section introduces a holography inverse-design problem for producing an image in depth along the direction of wave propagation.
- 3. Results.: The holography example designs an optical system to create an image in depth along the propagation direction.
3.1. Holography.
The holography example designs a permittivity map so a scattering slab produces a target transmitted-wave pattern, using hPINN networks with embedded boundary conditions. Forward verification and inverse-design experiments compare soft constraints, penalty methods, augmented Lagrangians, and conventional PDE solvers.
- Problem setup: The design optimizes the lens permittivity ε in a rectangular electromagnetic domain to match a target transmitted intensity in the upper region.The target is a black-square pattern, while the electric field must satisfy the holography PDEs and boundary conditions.
- hPINN architecture: Three networks approximate ℜ[E], ℑ[E], and ε, while periodic and Dirichlet boundary conditions are embedded directly into the architecture.The permittivity bound ε ∈ [1, 12] is enforced through ε(x, y) = 1 + 11 sigmoid(N3(x, y)).
- Forward verification: When the training loss is ≲10^-4, the forward-problem L2 relative error is < 1% against the FDFD reference.The hPINN field agrees with FDFD, and the pointwise PDE-informed loss lies between −5 × 10^-3 and 5 × 10^-3.
- Soft constraints: The soft-constraint method is sensitive to µF: µF = 0.1 poorly satisfies the PDE, whereas µF = 10 satisfies it but yields an ill-conditioned, poor design.The smallest reported soft-constraint objective is ≈0.0547 at µF = 2, but the predicted field remains inaccurate.
- Augmented Lagrangian method: The augmented Lagrangian method keeps the PDE loss below 10^-4, achieves a 1.2% field error against FDFD, and resolves the penalty method’s convergence issue.Its objective does not worsen even when µF exceeds 10^3.
- Comparison with conventional methods: hPINN achieves objective values nearly matching FDFD and FEM, although the resulting designs differ because the holography problem has many local optima.The reported conventional-solver objectives are 0.0523 for FDFD and 0.0528 for FEM.
3.2. Fluids in Stokes flow.
The Stokes-flow topology optimization uses hPINN to represent velocity, pressure, and material density while enforcing PDE, boundary, and volume constraints. The augmented Lagrangian refinement produces a nearly exact fluid volume, improves PDE agreement, and matches the conventional objective.
- Problem setup: The design domain assigns ρ = 0 to solid and ρ = 1 to fluid, while intermediate densities make large-scale topology optimization tractable.The objective minimizes dissipated power under a fluid-volume constraint with γ = 0.9.
- Network and constraints: The hPINN represents u, v, p, and ρ with neural networks and embeds Dirichlet boundary conditions directly into the architecture.The density is clipped to [0, 1], with boundary and center restrictions used to avoid undesirable local minima.
- Soft constraints: The soft-constraint design reaches objective 13.73 versus 14.07 from FEM, but its 0.906 fluid volume violates the target γ = 0.9 and partly explains the lower objective.The authors report that the soft-constraint objective is smaller because the design slightly violates the volume constraint.
- Augmented Lagrangian method: The augmented Lagrangian method requires only ∼400 additional iterations, or ∼2% more computational cost, after the soft-constraint solution already satisfies the constraints well.This refinement is designed to improve constraint satisfaction without restarting the optimization.
- Augmented Lagrangian method: The refined design has fluid volume 0.901, L2 relative velocity error 2.0%, and objective 14.12, consistent with the conventional result.The fluid-volume loss decreases from 10^-5 to 10^-8 after augmented-Lagrangian training.
4. Conclusion.
hPINN achieves good constrained designs across holography and Stokes-flow examples, matching conventional objectives while producing simpler, smoother designs for non-unique solutions. The augmented Lagrangian method imposes hard constraints, but its convergence lacks a theoretical explanation and its pointwise multipliers may be costly at scale.
- hPINN was demonstrated on holography in optics and Stokes flow.
- The augmented Lagrangian method imposes hard constraints and achieves better designs than penalty methods when large penalties impair convergence.
- hPINN matches conventional PDE-constrained optimization objectives while producing simpler and smoother designs for non-unique solutions.
- hPINN does not rely on a numerical solver, supporting a one-size-fit-all inverse-design implementation.
- The augmented Lagrangian method lacks a theoretical convergence understanding, and one multiplier per residual location may be expensive for large-scale problems.
Appendix A. PDEs of the holography problem.
The appendix identifies the real and imaginary parts of Eq. (3.3), but the supplied passage does not include their expressions or further interpretation.
- The real and imaginary parts of Eq. (3.3) are introduced.
- The passage provides no displayed expressions for the real and imaginary parts.
- No additional PDE role or symbol definitions are stated in the supplied passage.
Appendix B. Solving holography problem via the finite-difference frequency-domain method.
The finite-difference frequency-domain baseline optimizes permittivity against a target pattern using PDE constraints and adjoint gradients. The design region is discretized at specified wavelength resolutions and optimized from a uniform vacuum-permittivity initialization.
- The method optimizes permittivity in the design volume to match a targeted pattern.
- PDE constraints and the adjoint method are computed with a finite-difference frequency solver.
- The computational domain uses 40 pixels per vacuum wavelength, or about 10 pixels per wavelength at permittivity 12.
- Permittivity ranges from 1 to 12, with a uniform unit-permittivity initial guess; optimization converges in about 5000 steps at relative tolerance 10^-8.
Appendix C. Solving holography problem via the finite element method.
The finite-element baseline uses region-dependent meshes and Gridap.jl to solve the PDEs. Topology optimization parameterizes permittivity through pixel variables, smooths them with a Helmholtz filter, and updates them using adjoint gradients and MMA.
- Finite-element meshes use different resolutions across the permittivity-one domain, design domain, and PML.
- The domain is meshed with GMSH and the PDEs are solved via Gridap.jl.
- Permittivity is parameterized as ϵ = 1 + 11p, with p in [0, 1].
- Design parameters are smoothed by a Helmholtz-type filter, differentiated with the adjoint method, and optimized using MMA.
Appendix D. Smoothed profile method for Stokes flow.
The smoothed profile method represents immersed solids through a smoothly varying profile derived from inverse permeability and advances pressure and velocity through staged updates. Its construction automatically fulfills no-slip and no-penetration conditions, using a specified computational discretization for steady-state simulations.
- Immersed-body representation: The smoothed profile method represents the immersed solid body with a hyperbolic tangent function whose profile is derived from the inverse permeability parameter.The profile equals 1 inside the solid, 0 in the fluid, and varies smoothly across the interface.
- Stokes-flow solution: The fluid flow is governed by the Stokes equations, with intermediate pressure and velocity fields computed in successive stages.The staged procedure includes pressure boundary conditions, intermediate velocity computation, and immersed-body velocity updates.
- Immersed-body representation: The method uses a prescribed immersed-body velocity of Vs = 0 in the simulations.This velocity is used when defining the immersed solid-body profile.
- Stokes-flow solution: The total pressure is updated by adding the extra pressure field to the intermediate pressure, pn+1 = p∗+ pp.The extra pressure field accounts for the immersed solid body.
- Boundary conditions: The staged equations automatically fulfill no-slip and no-penetration boundary conditions.This property is stated for the sequence of equations used by the smoothed profile method.
- Simulation setup: Simulations use a [0, 1] × [0, 1] domain with 50 × 50 quadrilateral elements, second-order spectral elements, and ∆t = 10−4 until steady state.The computational setup integrates the governing equations until steady state.