Source-linked AI summary
Machine Learning Inverse Problem for Topological Photonics
Laura Pilozzi, Francis A. Farrelly, Giulia Marcucci, Claudio Conti
TL;DR
Topological design requires effective methods for inverse problems in complex structures. This paper uses supervised neural-network regression with self-consistency and domain-specific networks to identify physically relevant solutions, yielding a general approach scalable to hundreds of topological devices.
Problem
The inverse problem in topological design is to engineer topology for complex structures while ruling out unphysical solutions.
Method
The method uses supervised machine-learning regression with a self-consistent procedure, categorical features, and separate neural networks for discontinuous domains and discrete variables.
Results
The approach successfully enables tailored engineering of protected edge states and is scalable to complex structures involving hundreds of topological devices.
Takeaways & Limitations
The general method can extend beyond the example considered to other physical systems in topological science, including polaritonics, quantum technologies, and ultracold atoms.
Abstract
from arXiv · showhide
Topological concepts open many new horizons for photonic devices, from integrated optics to lasers. The complexity of large scale topological devices asks for an effective solution of the inverse problem: how best to engineer the topology for a specific application? We introduce a novel machine learning approach to the topological inverse problem. We train a neural network system with the band structure of the Aubry-Andre-Harper model and then adopt the network for solving the inverse problem. Our application is able to identify the parameters of a complex topological insulator in order to obtain protected edge states at target frequencies. One challenging aspect is handling the multivalued branches of the direct problem and discarding unphysical solutions. We overcome this problem by adopting a self-consistent method to only select physically relevant solutions. We demonstrate our technique in a realistic topological laser design and by resorting to the widely available open-source TensorFlow library. Our results are general and scalable to thousands of topological components. This new inverse design technique based on machine learning potentially extends the applications of topological photonics, for example, to frequency combs, quantum sources, neuromorphic computing and metrology.
DISCUSSION
The paper solves topological photonic inverse design with supervised neural-network regression and a self-consistent check that excludes unphysical solutions. The approach addresses multivalued and discontinuous behavior while supporting protected edge-state engineering.
- DISCUSSION: Supervised regression combined with a self-consistent cycle rules out unphysical inverse-design solutions and enables tailored protected edge states.Tentative inverse-network solutions are evaluated through the direct-problem network before acceptance.
- DISCUSSION: Multivalued functions are handled with categorical trend features, while discontinuous domains use independent neural networks specialized to each domain.The trend labels distinguish positive and negative-gradient branches; domain-specific networks address discontinuities.
- DISCUSSION: The authors state that the approach can scale to complex structures containing hundreds of topological devices and extend to additional topological systems and applications.Suggested extensions include frequency-comb generation, quantum sources, 2D and 3D systems, and non-Hermitian systems.
- DISCUSSION: The method reconstructs edge-state dispersions for both direct and inverse problems across the full range of interest.Figure 4 compares the direct-problem reconstruction with the inverse-problem solution.
METHODS
The method combines transfer-matrix analysis of a periodic dielectric structure with neural networks that identify topological edge states and accommodate discontinuities and mixed variable types.
- Neural-network implementation: The neural-network implementation uses TensorFlow, separates data into two networks across the ξ = 0 discontinuity, and flexibly handles continuous and discrete variables.Training uses randomly split training and validation sets, with iterative updates until validation error converges.
- Transfer-matrix model: The transfer matrix is constructed from interface and layer-propagation matrices to model light propagation through the periodic dielectric structure.The structure has period D = qd0, with normalized B-layer thicknesses used in the propagation description.
- Band and gap identification: Bulk bands and gaps are located from transfer-matrix eigenvalues or the reflection coefficient, with bands satisfying ρ^2 ≤ 1 and gaps satisfying ρ^2 > 1.The eigenvalue e^ik(ω)D provides an alternative route for locating system gaps.
- Edge-state conditions: Edge states are determined by boundary conditions, transfer-matrix eigenvectors, and zeros of a function derived from their proportionality condition.The boundary vector and eigenvectors yield det(b1, v±) = 0 in gaps where |λ±| < 1.
- Edge-state conditions: The function Q identifies allowed edge states in gaps where |ρ| > 1 and Q·ρ > 0, while the edge-mode count equals the modulus of the associated winding-number invariant.Edge states cannot occur in gaps where Q does not change sign.