Scalable multitask Gaussian processes for complex mechanical systems with functional covariates

Abstract: 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.