Li Liao, Pengfei Shen, Yifan Peng
Eigenanalysis of partial differential operators is central to reduced-order physical simulation, but neural shape-space eigenanalysis has largely been limited to natural Neumann boundary conditions. This prevents direct control over supports, openings, heat-exchange boundaries, and other boundary effects that change the underlying operator. We extend neural eigenanalysis for Laplace-type operators to Dirichlet, Robin, and mixed boundary conditions. Boundary placement and Robin coefficients are treated as model inputs, giving a joint shape-boundary space where eigenfunctions and spectra vary continuously with both geometry and boundary configuration. The resulting boundary-aware bases support resonance tuning, reduced-order elastic simulation with changing supports, and transient heat analysis under controllable boundary exchange.