Handling noise and overfitting in surrogate models based on non-uniform rational basis spline entities

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.