This paper proposes a general and systematic approach to generate a surrogate model based on non-uniform rational basis spline (NURBS) hyper-surfaces as a solution of a constrained non-linear programming problem, which is solved through a gradient-based algorithm by considering two formulations. The first formulation, which is the general one, is prone to the curse of dimensionality. The second one separates dimensions in the NURBS domain and allows saving computational time and resources. The optimisation process consists of three steps, which are solved sequentially. Each step is characterised by different design variables and solving strategies. The gradient of the cost function, for both formulations and for each optimisation step, is derived analytically by exploiting the properties of NURBS entities, hence enhancing computational efficiency and accuracy. One of the main features of the presented methodology lies the incorporation of a smoothing term in the problem formulation. This term plays a crucial role in controlling overfitting phenomenon and attenuating local oscillations within the dataset. These oscillations may be due to various sources, such as inherent noise, model instabilities, measurement uncertainties, or numerical precision limitations. The effectiveness of the proposed approach is tested on both analytical benchmark problems and real-world engineering problems taken from the literature.

中文翻译：

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基于非均匀有理基样条实体的代理模型中的噪声和过拟合处理

本文提出了一种通用且系统的方法来生成基于非均匀有理基样条（NURBS）超曲面的代理模型，作为约束非线性规划问题的解决方案，该问题通过基于梯度的算法来解决，考虑两种配方。第一个公式是一般公式，很容易受到维数灾难的影响。第二种方法在 NURBS 域中分离尺寸，从而节省计算时间和资源。优化过程包括三个步骤，依次求解。每个步骤都有不同的设计变量和解决策略。两种公式和每个优化步骤的成本函数梯度都是通过利用 NURBS 实体的属性来分析得出的，从而提高了计算效率和准确性。所提出方法的主要特征之一在于在问题表述中纳入平滑项。该术语在控制过度拟合现象和衰减数据集中的局部振荡方面发挥着至关重要的作用。这些振荡可能是由于各种来源造成的，例如固有噪声、模型不稳定性、测量不确定性或数值精度限制。所提出方法的有效性在分析基准问题和文献中的实际工程问题上进行了测试。