Boundary-aware Neural Model Reduction for PDEs

摘要

Eigenanalysis of partial differential operators is essential in reduced-order modeling for physical simulation, providing eigenmode representations in elasticity, acoustics, and transient heat transport. While recent neural, mesh-free, and geometry-agnostic approaches enable differentiable eigenanalysis over continuously parameterized shape spaces, boundary conditions of eigenfunctions are limited to Neumann-type (natural) conditions, hindering their applicability in scenarios where boundary conditions must be precisely controlled or optimized. In this work, we focus on Laplace-type operators and extend shape space neural eigenanalysis to handle boundary conditions beyond natural Neumann settings. Building on the same variational, energy-based formulation, we show that Dirichlet, Robin, and mixed boundary conditions can be incorporated without altering the underlying eigenvalue optimization principle. In our formulation, boundary configurations—including boundary placement and Robin coefficients modeling boundary exchange processes—are treated as first-class parameters rather than fixed constraints. When combined with shape-parameterized domains, this leads to a joint shape–boundary space formulation, allowing eigenfunctions and spectra to be evaluated consistently across variations in both geometry and boundary configuration. We conduct experiments on representative applications such as boundary-driven spectral optimization for rigid-walled cavity resonance tuning, reduced-order simulation with changing supports, and analysis of transient thermal behavior under varying boundary exchange conditions. By elevating boundary conditions from fixed constraints to operator-wise parameterization, our approach broadens the applicability of Laplace-type neural eigenanalysis to physical systems where boundary constraints serve as critical design and control variables.