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 1D 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/without minimum-height constraints -- we evaluate accuracy by ray-traced normalized mean absolute error (NMAE). Our neural network approach converges faster and achieves consistently lower NMAE than the deconvolution method, and handles height constraints naturally. We discuss how the method may be extended to rotationally symmetric and full three-dimensional settings via iterative correction schemes.

翻译：我们研究了设计二维反射器的逆问题，该反射器将有限扩展光源的光转换为指定的远场分布。我们提出了一种基于神经网络参数化反射器高度的方法，并开发了两个可微目标函数：(i) 直接变量替换损失，通过学习的逆映射推送光源分布；以及(ii) 基于网格的损失，将目标空间网格映射回光源，对交点进行积分，并在光源不连续时保持连续性。梯度通过自动微分获得，并使用稳健的拟牛顿法进行优化。作为对比，我们构建了一个基于简化有限光源近似的去卷积基线：从通量平衡中恢复一维单调映射，得到以积分因子形式求解的常微分方程；该求解器嵌入改进的Van Cittert迭代中，并采用非负性裁剪和光线追踪前向算子。在四个基准测试中——连续与非连续光源、以及有无最小高度约束——我们通过光线追踪归一化平均绝对误差评估精度。与去卷积方法相比，我们的神经网络方法收敛更快，始终获得更低的NMAE，并能自然处理高度约束。我们讨论了如何通过迭代校正方案将该方法扩展到旋转对称及完整三维设置。