Neural network methods for two-dimensional finite-source reflector design

Abstract

Abstract We address the inverse problem of designing two-dimensional reflectors that transform light from a finite, extended source into a prescribed far-field distribution. We propose a neural-network parameterization of the reflector height and develop two differentiable objective functions: (i) a direct change-of-variables loss that pushes the source distribution through the learned inverse mapping, and (ii) a mesh-based loss that maps a target-space grid back to the source, integrates over intersections, and remains continuous even when the source is discontinuous. Gradients are obtained via automatic differentiation and optimized with a robust quasi-Newton method. As a comparison, we formulate a deconvolution baseline built on a simplified finite-source approximation: a one-dimensional monotone mapping is recovered from flux balance, yielding an ordinary differential equation solved in integrating-factor form; this solver is embedded in a modified Van~Cittert iteration with nonnegativity clipping and a ray-traced forward operator. Across four benchmarks---continuous and discontinuous sources, and with and without minimum-height constraints---we evaluate accuracy by the ray-traced normalized mean absolute error. Our neural-network approach converges faster and achieves consistently lower errors than the deconvolution method, and handles height constraints naturally. On our two main benchmarks the neural network reduces this error to $2\times 10^{-5}$ for a continuous source and $5\times 10^{-5}$ for a discontinuous source within a few seconds of wall-clock time on a single GPU, against $4\times 10^{-3}$ and $5\times 10^{-2}$ for the deconvolution baseline after several hundred seconds---two to three orders of magnitude lower error at two orders of magnitude less wall-clock time. We discuss how the method may be extended to rotationally symmetric and full three-dimensional settings via iterative correction schemes.