Coordinate-Specific Factor Design in Low-Rank Physics-Informed Solvers

Low-rank physics-informed solvers use products of one-dimensional factors, but coordinates can have distinct physical structures. Suitable factors can improve accuracy, yet coupled changes in family, capacity, and assignment complicate attribution. We introduce DimSpec-LR, a coordinate-specific framework with a shared CP interaction and a unified interface for neural, dictionary-based, and hybrid factors. Parameter-matched interventions examine family selection, capacity allocation, and coordinate assignment under controlled training configurations. Experiments across four PDE families demonstrate that physically aligned factors can substantially improve accuracy under a fixed training budget. On a boundary-layer problem, the prescribed profile reduces relative error from $0.8546$ to $1.59\times 10^{-3}$ compared with a generic homogeneous control. However, a richer homogeneous alternative achieves an even lower error of $4.04\times 10^{-4}$, revealing that coordinate suitability does not necessarily require heterogeneous factor families. Additional experiments show that allocating spectral capacity to cover the required temporal frequencies improves accuracy and that target-frequency coverage alone does not determine trained performance. A separate source-specified CDR problem shows that fixed local factors remain effective without constructing an analytic target solution. Initialization and preconditioning controls clarify their contributions to complete-solver accuracy. These findings establish a controlled evaluation methodology for physically informed factor design and highlight the importance of distinguishing coordinate suitability, capacity allocation, and architectural heterogeneity.