NOVA: Nullspace Dynamics in Neural Basis Solvers for Moving Boundary PDEs, Transport, Rank Transitions, and Reprojection

Multiphysics coupling with moving boundaries poses a major challenge to neural solvers, as evolving interfaces induce time dependent constraints. Existing boundary treatments face an inherent trade-off between constraint accuracy and update cost. To address this challenge, we propose NOVA (**N**ullspace Dynamics in Neural Basis S**O**lvers for Mo**V**ing Bound**A**ry PDEs, Transport, Rank Transitions, and Reprojection). The method explicitly decomposes neural basis coefficients into a boundary lift and a component parameterized by instantaneous nullspace coordinates, then reprojects the previous physical state onto the updated nullspace. This construction avoids boundary parameter sweeps by eliminating penalty weights and stabilization parameters. To avoid costly singular value decompositions (SVDs) at every time step, we derive horizontal transport equations for the nullspace basis on the Stiefel manifold and develop an accompanying algorithm. Furthermore, spectral analysis establishes a rank saturation criterion for boundary collocation. Across eight multiphysics benchmarks covering phase change and flow, multiphase flow, and free boundary problems, NOVA achieves best or competitive field accuracy with substantially lower boundary errors than baselines, providing a new paradigm for physics-informed neural networks based on features and mesh-free representations to efficiently handle complex moving geometries. Our code is available [here](https://anonymous.4open.science/r/NOVA-3C44).