Numerical Methods for Partial Differential Equations Seminar

PDE-constrained optimization (PDE-CO) is central to scientific and engineering design but remains limited by computationally expensive solvers and the lack of reusable structure across tasks. Solver-informed learning offers a promising alternative: training AI models in the input space of PDE solvers to preserve physics fidelity, yield interpretable parameterizations, and support generalization while reducing overall design cost. This presentation outlines how input-space representations can bridge physics, optimization, and machine learning—encoding mathematical structure, enabling effective use of low-fidelity solvers, and revealing latent spaces suitable for optimization.

The magnetic Schrödinger equation provides a (probabilistic) model of the motion of a charged particle in an electromagnetic field. The associated (time-independent) eigenvalue problem provides probability densities, via normalized eigenvectors, of the location of the charged particle at certain energies associated with the eigenvalues. Even for magnetic fields that have a seemingly simple structure, there appear to be non-trivial computational challenges for approximating eigenvalues and eigenvectors, particularly as the strength of the magnetic field increases. We will highlight some of these challenges, and introduce two independent approaches that aim to address these challenges in two different regimes. We will also provide some background concerning what is currently well-understood about the problem (not as much as one might hope), and hint at some progress toward improvements on the theoretical and computational fronts.