I develop principled algorithms for PDEs that are difficult for existing methods, such as equations with rough forcing and functional PDEs posed on infinite-dimensional spaces. I also develop convergence guarantees for these machine-learning-based approaches.
Relevant research
Solving Roughly Forced Nonlinear PDEs via Misspecified Kernel Methods and Neural Networks
R. Baptista, E. Calvello, M. Darcy, and 3 more authors
@article{baptista2026rough,title={Solving Roughly Forced Nonlinear {PDEs} via Misspecified Kernel Methods and Neural Networks},author={Baptista, R. and Calvello, E. and Darcy, M. and Owhadi, H. and Stuart, A. M. and Yang, X.},journal={Mathematics of Computation},year={2026},doi={10.1090/mcom/4219},}
Solving Functional PDEs with Gaussian Processes and Applications to Functional Renormalization Group Equations
@article{yang2025functional,title={Solving Functional {PDEs} with {G}aussian Processes and Applications to Functional Renormalization Group Equations},author={Yang, X. and Darcy, M. and Hudes, M. and Alexander, F. J. and Eyink, G. and Owhadi, H.},journal={Journal of Computational Physics},volume={563},pages={115085},year={2026},doi={10.1016/j.jcp.2026.115085},}
