Functional covariates arise in many scientific and engineering applications when model inputs take the form of time-dependent or spatially distributed profiles, such as varying boundary conditions or changing material behaviours. In addition, new practices in digital simulation require predictions accompanied by confidence intervals. Models based on Gaussian processes (GPs) provide principled uncertainty quantification. However, GPs capable of jointly handling functional covariates and multiple correlated functional tasks remain largely under-explored. In this work, we extend the framework of GPs with functional covariates to multitask problems by introducing a fully separable kernel structure that captures dependencies across tasks and functional inputs. By taking advantage of the Kronecker structure of the covariance matrix, the model is made scalable. The proposed model is validated on a synthetic benchmark and applied to a realistic structure, a riveted assembly with functional descriptions of the material behaviour and response forces. The proposed functional multitask GP significantly improves over single task GPs. For the riveted assembly, it requires less than 100 samples to produce an accurate mean and confidence interval prediction. Despite its larger number of parameters, the multitask GP is computationally easier to learn than its single task pendant.

翻译：功能协变量广泛存在于科学与工程应用中，当模型输入呈现为时间依赖或空间分布的轮廓时（例如变化的边界条件或时变材料行为）。此外，数字仿真领域的新实践要求预测结果需附带置信区间。基于高斯过程的模型能够提供理论完备的不确定性量化。然而，能够同时处理功能协变量与多重相关功能任务的高斯过程模型仍鲜有研究。本文通过引入完全可分离的核结构（该结构能够捕捉任务间及功能输入间的依赖关系），将面向功能协变量的高斯过程框架扩展至多任务问题。借助协方差矩阵的克罗内克结构特性，所提模型实现了可扩展性。该模型在合成基准测试中得到验证，并应用于一个真实结构——铆接装配体（其材料行为与响应力均以功能形式描述）。所提出的功能多任务高斯过程较单任务模型有显著提升。对于铆接装配体案例，仅需不足100个样本即可生成精确的均值与置信区间预测。尽管参数数量更多，该多任务高斯过程在计算学习效率上反而优于对应的单任务模型。