Abstract:

Gaussian process regression is a widely used Bayesian framework for non-parametric function inference and learning in abstract vector spaces. However, the computational complexity of standard inference algorithms typically scales cubically with the size of the training dataset, limiting its applicability to large-scale problems. This limitation is further exacerbated in physics-informed machine learning, where physical constraints, such as linear operator constraints, are enforced. Such constraints may come from differential and integral operators. 
 
This project will develop scalable algorithms for constrained Gaussian process regression, thereby expanding its application in various scientific domains as a principled framework for uncertainty quantification. By leveraging randomized numerical linear algebra, preconditioning techniques, and hierarchical matrix methods, we aim to design efficient algorithms that incorporate constraints while maintaining computational scalability. The research will establish theoretical properties of constrained Gaussian processes, develop efficient numerical solvers to address computational bottlenecks, and implement these methods for inverse problems and structured learning tasks. 
 
To motivate the development of these algorithms, we will consider concrete inverse problems from diverse scientific domains, including pharmacometrics and medical imaging, which present fundamentally different mathematical structures and constraints. By applying our methods across these fields, we will demonstrate the versatility and robustness of scalable constrained Gaussian process regression in real-world applications. The outcomes of this research will contribute to the broader adoption of Gaussian processes as an efficient and uncertainty-aware modeling tool, enabling scalable and physically consistent predictions in complex scientific and engineering problems.