Deep Gaussian Processes for Functional Maps
This addresses function-on-function regression for applications like spatiotemporal forecasting and climate modeling, but it is incremental as it builds on existing GP and deep learning methods.
The paper tackled the problem of learning mappings between functional spaces under noisy, sparse, and irregularly sampled data by proposing Deep Gaussian Processes for Functional Maps (DGPFM), which achieved improved predictive performance and uncertainty calibration on real-world and PDE benchmark datasets.
Learning mappings between functional spaces, also known as function-on-function regression, plays a crucial role in functional data analysis and has broad applications, e.g. spatiotemporal forecasting, curve prediction, and climate modeling. Existing approaches, such as functional linear models and neural operators, either fall short of capturing complex nonlinearities or lack reliable uncertainty quantification under noisy, sparse, and irregularly sampled data. To address these issues, we propose Deep Gaussian Processes for Functional Maps (DGPFM). Our method designs a sequence of GP-based linear and nonlinear transformations, leveraging integral transforms of kernels, GP interpolation, and nonlinear activations sampled from GPs. A key insight simplifies implementation: under fixed locations, discrete approximations of kernel integral transforms collapse into direct functional integral transforms, enabling flexible incorporation of various integral transform designs. To achieve scalable probabilistic inference, we use inducing points and whitening transformations to develop a variational learning algorithm. Empirical results on real-world and PDE benchmark datasets demonstrate that the advantage of DGPFM in both predictive performance and uncertainty calibration.