On Parameter Decoupling, Conditioning, and Interface Transmission in Multi-Objective Physics-Informed Neural Networks

Multi-objective formulations in Physics-Informed Neural Networks (PINNs) routinely exhibit severe gradient conflict between boundary constraints, interface transmission conditions, and interior differential residuals. While heuristic gradient surgery and dynamic loss weighting are commonly employed, the fundamental mechanics governing parameter decoupling, spatial basis localization, and optimization conditioning remain uncharacterized. In this work, we resolve the duality between parameter subspace decoupling and spatial basis localization. To establish theoretical limits of conflict elimination, we construct an exact algebraic parameter decomposition equipped with orthogonal readout projections that guarantees pairwise gradient orthogonality ($\\cos\\rho = 0$) on saturated spatial partitions, blended across geometric interfaces via a $C^2$ Quintic Hermite routing gate. Across diverse computational mechanics benchmarks under strictly matched parameter capacity ($P = 9{,}216$), we demonstrate that spatial basis localization—rather than parameter decoupling—is the primary driver of boundary stiffness mitigation. While exact parameter decoupling reduces boundary interface loss by $6.4\\times$ relative to global coordinate MLPs, a standard localized cubic B-spline representation matches boundary fidelity ($2.08 \\times 10^{-4}$ vs.\\ $2.15 \\times 10^{-4}$) while delivering $1.53\\times$ lower interior error ($0.257\\,\\mathrm{T}$ vs.\\ $0.392\\,\\mathrm{T}$) without parameter partitioning. Furthermore, through closed-form spectral derivation of the parameter-sharing metric tensor $\\mathbf{H}_\\alpha \\mathbf{H}_\\alpha^T$, we prove that the representational rank is strictly invariant ($\\mathrm{rank}=2$ for all $\\alpha < 1$); continuous parameter sharing operates purely as an anisotropic optimizer preconditioner in finite-budget optimization. Finally, our Sobolev analysis proves why single smooth trial spaces fail to satisfy non-zero interface flux jumps, providing a principled representation selection framework for scientific computing.