Physics-informed neural networks with hard constraints for inverse design

By Lu Lu, Raphaël Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, Steven G. Johnson. SIAM Journal on Scientific Computing, 2021

806 citations recorded by the source corpus.

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.

Subfields: Civil and Structural Engineering, Statistical and Nonlinear Physics, Structural Biology

View Paper: https://www.semanticscholar.org/paper/fef2135b3ae7b27ab28ddf41a943bd2ddc5d5113

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Citations in the corpus

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “DeepXDE is used by the authors for: for inverse scattering (Chen et al, 2020b), or deriving mechanical characteristics from materials with MFNNs (Lu et al, 2020).”
Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems (2021, methodology): “Neural network architectures can also be modiﬁed to satisfy automatically and exactly the required Dirichlet boundary conditions [20, 21, 22], Neumann boundary conditions [23, 24], Robin boundary conditions [25], periodic boundary conditions [26, 8], and interface conditions [25].”
MIONet: Learning multiple-input operators via tensor product (2022, methodology): “As an active area of research in SciML, different methods have been developed to solve ordinary and partial differential equations (ODEs and PDEs) by parameterizing the solutions via neural networks (NNs), such as physics-informed NNs (PINNs) [48, 30, 35, 37, 47], deep Ritz method [43], and deep Gal”
Multi-scale modeling in thermal conductivity of Polyurethane incorporated with Phase Change Materials using Physics-Informed Neural Networks (2023, background): “PINNs offer the advantage of integrating physics-based knowledge of the system with data-driven learning, making them well-suited for solving inverse problems [33] and handling complex multi-scale phenomena.”
Characterizing possible failure modes in physics-informed neural networks (2021, background): “Previous work has tried to impose hard constraints in ML (both within the context of SciML and otherwise) [6, 21, 22, 24, 40], although this can be computationally expensive, and does not guarantee better results or convergence.”
Physics-informed neural networks (PINNs) for fluid mechanics: a review (2021, methodology): “, using multi-fidelity framework [34], adaptive activation functions [35] and dynamic weights of the loss function [36], hard constraints [37] and CNN-based network architectures [38], which can improve the performance of PINNs on different problems.”
Analyses of internal structures and defects in materials using physics-informed neural networks (2022, background): “For these problems, PINNs can incorporate the design target as a loss term, aspects of which have been preliminarily explored in (60).”
A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data (2021, methodology): “Here, we ﬁrst introduce how to enforce periodic BCs in neural networks in 1D [13, 49] by constructing special feature expansion as discussed in Section 3.1.2, and then discuss how to extend it to 2D.”