Many data-generating processes arising in physics and engineering possess intrinsic geometric structures, such as symmetries, conservation laws, variational principles, and differential equation constraints. Incorporating these structures into machine learning models is crucial for achieving physically meaningful and reliable predictions.

In this talk, I will introduce a structure-preserving kernel-based learning framework for recovering unknown functions while respecting the underlying geometric properties of the problem. The proposed method admits a closed-form solution, achieves strong numerical performance, and often outperforms existing approaches. In the manifold setting, the learned solution is globally defined and independent of the choice of local coordinates. I will also present theoretical guarantees, including convergence results under both fixed and adaptive regularization schemes, and discuss applications to structured learning problems arising in scientific computing.