# Physics-informed neural networks with hard constraints for inverse design

@article{Lu2021PhysicsinformedNN, title={Physics-informed neural networks with hard constraints for inverse design}, author={Lu Lu and Rapha{\"e}l Pestourie and Wenjie Yao and Zhicheng Wang and Francesc Verdugo and Steven G. Johnson}, journal={SIAM J. Sci. Comput.}, year={2021}, volume={43}, pages={B1105-B1132} }

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…

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## 23 Citations

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## References

SHOWING 1-10 OF 53 REFERENCES

Physics-informed machine learning

- 2021

Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into…

Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

- Mathematics, PhysicsJ. Comput. Phys.
- 2019

A new method is proposed with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty, which can be readily applied to other types of stochastic PDEs in multi-dimensions.

Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

- Computer ScienceJ. Comput. Phys.
- 2019

Abstract We introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear…

DeepXDE: A Deep Learning Library for Solving Differential Equations

- Computer Science, PhysicsAAAI Spring Symposium: MLPS
- 2020

An overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation, and a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs.

Simulator-based training of generative neural networks for the inverse design of metasurfaces

- Physics, Computer ScienceNanophotonics
- 2019

This work presents a new type of population-based global optimization algorithm for metasurfaces that is enabled by the training of a generative neural network and observes that the distribution of devices generated by the network continuously shifts towards high performance design space regions over the course of optimization.

Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks

- Computer Science, MathematicsSIAM J. Sci. Comput.
- 2020

Two new Physics-Informed Neural Networks (PINNs) are proposed for solving time-dependent SPDEs, namely the NN-DO/BO methods, which incorporate the DO/BO constraints into the loss function with an implicit form instead of generating explicit expressions for the temporal derivatives of the Do/BO modes.

Physics-informed neural networks for inverse problems in nano-optics and metamaterials.

- Physics, MedicineOptics express
- 2020

The emerging paradigm of physics-informed neural networks (PINNs) are employed for the solution of representative inverse scattering problems in photonic metamaterials and nano-optics technologies and successfully apply mesh-free PINNs to the difficult task of retrieving the effective permittivity parameters of a number of finite-size scattering systems.

Deep learning enabled inverse design in nanophotonics

- Computer Science
- 2020

The recent progress in the application of deep learning to the inverse design of nanophotonic devices is discussed, mainly focusing on the three existing learning paradigms of supervised-, unsupervised-, and reinforcement learning.

Topology Optimization Accelerated by Deep Learning

- Computer ScienceIEEE Transactions on Magnetics
- 2019

It is numerically shown that the computational cost for the topology optimization can be reduced without the loss of optimization quality.

Multiscale topology optimization using neural network surrogate models

- Computer ScienceComputer Methods in Applied Mechanics and Engineering
- 2019

Because the derivative of the surrogate model is important for sensitivity analysis of the macroscale topology optimization, a neural network training procedure based on the Sobolev norm is described, and an alternative method is developed to enable creation of void regions.