We present a method for fitting monotone curves using cubic B-splines with a monotonicity constraint on the coefficients. We explore different ways of enforcing this constraint and analyze their theoretical and empirical properties. We propose two algorithms for solving the spline fitting problem: one that uses standard optimization techniques and one that trains a Multi-Layer Perceptrons (MLP) generator to approximate the solutions under various settings and perturbations. The generator approach can speed up the fitting process when we need to solve the problem repeatedly, such as when constructing confidence bands using bootstrap. We evaluate our method against several existing methods, some of which do not use the monotonicity constraint, on some monotone curves with varying noise levels. We demonstrate that our method outperforms the other methods, especially in high-noise scenarios. We also apply our method to analyze the polarization-hole phenomenon during star formation in astrophysics. The source code is accessible at \texttt{\url{https://github.com/szcf-weiya/MonotoneSplines.jl}}.

翻译：我们提出了一种使用三次B样条拟合单调曲线的方法，并对系数施加单调性约束。我们探索了实施该约束的不同方式，并分析了其理论与实证性质。我们提出了两种求解样条拟合问题的算法：一种采用标准优化技术，另一种训练多层感知机（MLP）生成器以近似不同设置和扰动下的解。当需要重复求解该问题（例如使用自助法构建置信带）时，生成器方法能够加速拟合过程。我们将所提方法与若干现有方法（包括部分未使用单调性约束的方法）在具有不同噪声水平的单调曲线上进行了评估。结果表明，我们的方法在性能上优于其他方法，尤其是在高噪声场景下。我们还将该方法应用于天体物理中恒星形成过程中的极化凹陷现象分析。源代码见 \texttt{\url{https://github.com/szcf-weiya/MonotoneSplines.jl}}。