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

Yuyao Chen; Lu Lu; George Em Karniadakis; Luca Dal Negro

Source-linked AI summary

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

Yuyao Chen, Lu Lu, George Em Karniadakis, Luca Dal Negro

arXiv:1912.01085v2Dec 2, 2019physics.comp-phphysics.optics

TL;DR

Inverse scattering in systems with strong multiple scattering makes recovering effective material properties difficult. The paper applies physics-informed neural networks to reconstruct effective permittivity in finite-size scattering systems and validates the framework numerically.

Problem
Recovering material properties from inverse scattering is difficult when strong multiple light scattering complicates inversion of physics-driven differential models.
Method
The paper uses mesh-free physics-informed neural networks that constrain surrogate solutions by enforcing the governing partial differential equations.
Results
The framework successfully reconstructs effective permittivity in representative finite-size scattering arrays and is validated through numerical simulations.
Takeaways & Limitations
PINNs provide a validated framework for effective-medium reconstruction that incorporates finite-size and radiation effects in scattering arrays.
Takeaways & Limitations
The retrieved permittivity distribution may differ substantially because the inverse problem does not always have a unique solution.

Abstract

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In this paper we employ the emerging paradigm of physics-informed neural networks (PINNs) for the solution of representative inverse scattering problems in photonic metamaterials and nano-optics technologies. In particular, we successfully apply mesh-free PINNs to the difficult task of retrieving the effective permittivity parameters of a number of finite-size scattering systems that involve many interacting nanostructures as well as multi-component nanoparticles. Our methodology is fully validated by numerical simulations based on the Finite Element Method (FEM). The development of physics-informed deep learning techniques for inverse scattering can enable the design of novel functional nanostructures and significantly broaden the design space of metamaterials by naturally accounting for radiation and finite-size effects beyond the limitations of traditional effective medium theories.

I. INTRODUCTION

The paper develops physics-informed neural networks (PINNs) for inverse electromagnetic scattering in nano-optics and metamaterials, addressing ill-posed, computationally difficult problems in complex multi-particle geometries. It demonstrates parameter retrieval, permittivity reconstruction, and inverse design validated by two-dimensional FEM simulations.

Motivation: Strong multiple scattering, nonlinearities, and noise make inverse scattering in complex multi-particle geometries intrinsically ill-posed and computationally intractable.Traditional numerical techniques therefore fail to predict desired system parameters.
Approach: The paper proposes PINNs for different inverse electromagnetic scattering problems directly relevant to nano-optics and metamaterials technologies.PINNs are presented as a general framework for solving forward and inverse partial differential equation problems.
Applications: PINNs retrieve effective-medium parameters for finite-size nanocylinder clusters arranged in periodic and aperiodic geometries.The paper specifically addresses effective medium determination and parameter retrieval.
Applications: The method reconstructs unknown objects’ spatial electric-permittivity distributions from synthetic scattering data and determines optimal dielectric permittivity for optical cloaking layers beyond the quasi-static limit.These demonstrations extend from spatial reconstruction to cloaking-layer design.
Validation and significance: All results are validated with two-dimensional Finite Element Method simulations, supporting inverse design of scattering nanostructures and photonic metamaterials from scattered fields.The framework also enables retrieval of unknown dielectric permittivity from near-field optical imaging data.

II. PHYSICS-INFORMED NEURAL NETWORKS

PINNs use neural networks as PDE-constrained surrogate models, restricting admissible solutions through physics, boundary conditions, initial conditions, and field observations. For inverse problems, unknown parameters are optimized jointly with network weights by minimizing an augmented loss, requiring only one training dataset and no inverse-parameter data.

Physics-informed constraint: PINNs restrict neural-network solutions by enforcing the governing partial differential equation rather than relying solely on standard deep-learning constraints.The method uses relatively simple feed-forward architectures and automatic differentiation to impose the PDE model.
Data requirements: PINNs require only one training dataset and no data on the inverse parameters, supporting unsupervised inverse scattering from measured or synthetic field data.These properties reduce the burden of massive datasets used by non-physics-constrained deep-learning approaches.
Forward and inverse problems: The only difference between forward and inverse PINNs is adding the Li loss term, which incurs an insignificant computational cost.PINNs are therefore presented as effective for highly nonlinear, dispersive, and ill-posed inverse problems.
Loss construction: The surrogate output is constrained to satisfy the PDE in the domain, boundary conditions, initial conditions, and available field observations.For inverse problems, an additional residual term compares the predicted and observed solution values.
Loss construction: The inverse PINN loss combines weighted PDE-residual, training-data, and initial or boundary-condition terms, then minimizes it jointly over network parameters and unknown physical parameters.The loss is L(θ, λ) = wfLf(θ, λ; Tf) + wiLi(θ, λ; Ti) + wbLb(θ, λ; Tb), with residual, training, and condition points defining the respective terms.

III. PINNS FOR THE HOMOGENIZATION OF FINITE-SIZE METAMATERIALS

This section formulates finite-size metamaterial homogenization as an inverse medium problem solved with PINNs constrained by the Helmholtz equation and trained on FEM scattering data. The method retrieves effective permittivity profiles reproducing the original fields, including non-homogeneous and lossy responses beyond traditional effective medium theory.

Method: PINNs retrieve effective permittivity profiles for finite-size dielectric metamaterials by training on synthetic scattering data generated with FEM.The retrieved profile is validated through a forward FEM simulation and comparison of total electric-field distributions using the L2 error norm.
Strong-scattering regime: Under stronger scattering, PINNs retrieve a non-homogeneous effective medium containing a spatial region of negative effective permittivity, with an L2 error of 5%.The resonant permittivity profile accounts for scattering and radiation effects beyond traditional effective medium theory.
Non-periodic arrays: The same PINN procedure applies to non-periodic arrays, retrieving asymmetric effective-permittivity distributions with negative-permittivity regions and an L2 error of 3.8%.For the Vogel spiral, the negative-permittivity region effectively accounts for strong radiation effects.
Radiation losses: The framework generalizes to complex effective permittivity, retrieving radiation losses through Im{εr(x, y)} with maximum values of 10^-4, 0.6, and 0.3 across three cases.The resulting effective-permittivity values remain within a total 3% error of previously obtained values.

IV. PINN FOR INVERSE MIE SCATTERING

This section demonstrates that PINNs can retrieve optical parameters and internal fields of single, coated, and multi-object nanostructures from surrounding or external field information constrained by physical PDE models. The results include low field-reconstruction errors, while multi-object retrieval can be nonunique despite reproducing the same scattering behavior.

Single nanocylinder: PINNs retrieve a nanocylinder’s permittivity and internal electric field using only its external field and a physical PDE model.The trainable permittivity is inferred from analytically computed external-field data while the internal field is reconstructed.
Single nanocylinder: 0.51% L2 error separates the PINN-reconstructed nanocylinder field from the analytical Mie-theory field.The comparison uses the reconstructed field and the real field distribution from Mie theory.
Coated nanocylinder: 1% L2 error is obtained for the reconstructed coated-nanocylinder field relative to the reference field.The reconstructed field follows training for 10^4 steps.
Coated nanocylinder: PINNs extend to coated nanocylinders by jointly retrieving the inner-core and coated-layer permittivities from total-field information.The host permittivity is fixed at ε0 = 1, and the retrieved parameters converge to the exact values used to generate the training data.
Multiple objects: For an asymmetric dimer, PINNs retrieve a permittivity profile whose two localized object permittivities qualitatively correspond to the input target.The retrieved profile is coupled to a PDE homogenization model and produces a FEM field with an L2 error of 0.3%.
Multiple objects: The multi-object inverse solution may be nonunique: a retrieved permittivity distribution can differ substantially from the input target while sharing its scattering behavior.This limitation means matching the field does not necessarily identify the original permittivity distribution uniquely.

V. PINN FOR INVISIBLE CLOAKING DESIGN

PINNs reformulate invisible cloaking as a parameter-retrieval problem, accurately recovering coating properties and reproducing undisturbed propagation in the small-particle regime. For wavelength-scale nanocylinders, PINNs discover spatially varying dielectric coatings that substantially reduce scattering, achieving a 75% reduction in scattering efficiency.

Small-particle invisible cloaking: The PINN-reconstructed electric field propagates through the coated nanocylinder without perturbation.Training data were generated from the Mie-theory field distribution for a coating satisfying the cloaking equation.
Small-particle invisible cloaking: PINNs accurately retrieve the coating-layer permittivity for a small coated nanocylinder, in complete agreement with the analytical solution.The PINN also retrieves magnetic relative permittivity equal to unity, indicating no magnetic response is needed for perfect scattering cancellation in this geometry.
Wavelength-scale radiation cloaking: For nanocylinders whose diameter equals the incoming wavelength, PINNs discover a spatially dependent coating permittivity that substantially reduces scattering despite the absence of perfect cloaking.The coating permittivity εc(x, y) is varied within the layer, and magnetic permeability is fixed to unity for a purely dielectric device.
Wavelength-scale radiation cloaking: 75% reduction in scattering efficiency is achieved by the PINN-identified coating layer under plane-wave excitation.Finite Element Method simulations confirm the reduced far-field scattering relative to the bare nanocylinder.
Wavelength-scale radiation cloaking: The framework retrieves material parameters from scattering data and designs dielectric devices with strongly reduced scattering properties.The stated framework applies to synthetic or measured scattering data.

VI. CONCLUSIONS

The paper introduces and validates PINNs for inverse scattering in photonic metamaterials and nano-optics, including effective-medium reconstruction, optical-parameter retrieval, and cloaking. FEM simulations validate the findings, while physics-constrained deep learning simplifies training and supports future design and homogenization advances.

VI. CONCLUSIONS: PINNs successfully solve representative inverse scattering problems in photonic metamaterials and nano-optics, including finite-size scattering arrays affected by radiation effects.The framework addresses deviations from the classical homogenization picture caused by finite-size and radiation effects.
VI. CONCLUSIONS: Physics-constrained deep-learning algorithms dramatically simplify data-training procedures compared with alternative machine-learning approaches.The approach leverages wave physics constraints during training.
VI. CONCLUSIONS: The PINNs framework retrieves multiple optical parameters from field distributions around nanomaterials, enabling direct access to material information from near-field imaging data.The method is fully validated using Finite Element Method (FEM) numerical simulations.
VI. CONCLUSIONS: 75% scattering abatement is achieved in dielectric coated cylinders with sizes comparable to the incoming-radiation wavelength.PINNs are applied to invisible cloaking with coated cylinders.
VI. CONCLUSIONS: Further PINNs development for inverse scattering and remote sensing could broaden design capabilities and advance homogenization theory with radiation effects.The current architectures rely on empirical tuning, while meta-learning may enable automated selection of optimum architectures.

FUNDING

The research was sponsored by the Army Research Laboratory under Cooperative Agreement Number W911NF-12-2-0023 and also supported by the DOE PhILMs project.

FUNDING: The work was also supported by the DOE PhILMs project (No. de-sc0019453).The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation.