[Paper Review] Physics-informed neural networks with hard constraints for inverse design

Lu Lu, Raphaël Pestourie+4 more|arXiv (Cornell University)|Feb 9, 2021

Topology Optimization in Engineering48 citations

TL;DR

The paper introduces hPINN, a physics-informed neural network framework that enforces hard constraints (via penalty and augmented Lagrangian methods) for PDE-constrained inverse design, demonstrated on holography and Stokes flow.

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.

Motivation & Objective

• Motivate and formulate PDE-constrained inverse design problems as topology optimization.
• Develop hPINN to solve inverse design without relying on traditional PDE solvers.
• Impose hard equality and inequality constraints within PINNs to ensure feasible designs.
• Compare hPINN performance with adjoint-based PDE-constrained optimization methods.

Proposed method

• Introduce two neural networks to approximate the solution u(x) and design parameter gamma(x).
• Construct PDE-informed loss by penalizing residuals of the PDEs and boundary conditions using automatic differentiation.
• Strictly impose Dirichlet and periodic boundary conditions by modifying network architecture (hard constraints).
• Present soft constraints, penalty method, and augmented Lagrangian method to enforce PDEs and inequality constraints as hard constraints.
• Provide algorithmic procedures (Algorithm 1 for penalty method, Algorithm 2 for augmented Lagrangian) for training.
• Demonstrate hPINN on holography in optics and a Stokes flow problem, comparing with traditional adjoint-based PDE-constrained optimization.

Experimental results

Research questions

• RQ1 Can hPINN enforce PDE constraints and hard inequality/boundary constraints effectively in inverse design problems?
• RQ2 How does hPINN perform on topology-optimized designs compared with conventional adjoint-based PDE solvers?
• RQ3 Do hard-constraint formulations lead to simpler or smoother designs in non-unique solution settings?
• RQ4 What are the convergence properties and practical training considerations for penalty vs augmented Lagrangian approaches?

Key findings

• hPINN achieves the same objective-function performance as conventional PDE-constrained optimization methods.
• Hard-constraint implementations often yield simpler and smoother designs without extra regularization on lengthscales or smoothness.
• Penalty and augmented Lagrangian methods address the ill-conditioning and convergence challenges of enforcing constraints in PINNs.
• Hard constraints via architectural modifications can enforce Dirichlet and periodic boundary conditions exactly.
• In forward tests, the PDE residuals can be driven to near zero, with L2 relative error improving as training progresses.