# Direct Optimization of a 3D Finite-Source Reflector via Neural-Network Parameterization

arXiv: https://arxiv.org/abs/2609.00899 · HTML: https://mlsanity.com/paper/2609.00899/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 38 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Linearly-Convergent Stochastic L-BFGS Algorithm](https://mlsanity.com/paper/1508.02087/) (2015)
- [Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?](https://mlsanity.com/paper/2405.04230/) (2024)
- [Gradient descent-based freeform optics design using algorithmic differentiable non-sequential ray tracing](https://mlsanity.com/paper/2302.12031/) (2023)
- [End-to-end Surface Optimization for Light Control](https://mlsanity.com/paper/2408.13117/) (2024)
- [A neural network approach for solving the Monge-Amp`ere equation with transport boundary condition](https://mlsanity.com/paper/2410.19496/) (2024)
- [Neural-network methods for two-dimensional finite-source reflector design](https://mlsanity.com/paper/2604.02184/) (2026)

## Abstract (arXiv)

We present a direct optimization method for three-dimensional freeform reflectors that transform the light of a finite-'etendue source into a prescribed far-field angular intensity distribution. The reflector profile is represented by a small neural network (a multilayer perceptron), which is trained end-to-end through a differentiable ray-tracing objective. We furthermore parameterize the emission directions in gnomonic coordinates, and show how we use this to ensure that every emitted ray intersects the reflector. At each iteration, the network is converted to a bicubic spline representation for ray-tracing efficiency, and intersections with this smooth surface are solved by a damped Newton solve, with gradients computed via the implicit function theorem. The traced output distribution is compared with the desired target on a 'soft' histogram, under an H-1-type spectral weighting that emphasizes long-range transport of flux to improve convergence. Optimization is performed using a BFGS method with self-scaled Broyden updates and a plateau-perturbation rule to prevent stalling. The method converges reliably within seconds on a single GPU for all examples tested.