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

PREPRINT / MANUSCRIPT UNDER SUBMISSIONTarget Venue: Journal of Computational Physics (JCP, Elsevier) KEY THEORETICAL & EMPIRICAL HIGHLIGHTS• Spatial Basis Localization vs. Parameter Decoupling: In controlled capacity-matched benchmarks ($\sim 9.2\mathrm{k}\text{--}10.4\mathrm{k}$ parameters across 8 seeds), spatial basis compactness—not parameter decoupling—is the primary passive mechanism relieving boundary stiffness. A standard unified cubic B-spline grid matches boundary compliance ($2.08 \times 10^{-4}$) while delivering $1.53\times$ lower interior error ($0.257\,\mathrm{T}$ vs. $0.392\,\mathrm{T}$) than decoupled grids without splitting parameter capacity.• Interface Transmission Mechanics & Flux Conservation: We prove analytically and verify numerically (1D composite diffusion with a $10\times$ conductivity jump) that single smooth trial spaces fundamentally cannot satisfy continuous normal flux $[\kappa \partial_n u] = 0$ without inducing an unphysical interface flux notch (68.7% flux error). Faithful physical flux transmission requires derivative kinks ($20/11$ vs. $2/11$) supported by material-branched representations (achieving 0.19% flux error).• Representation Rank Invariance & Preconditioning: We derive the closed-form latent alignment spectrum across continuous parameter sharing $\alpha \in [0, 1]$. We prove that representation rank is strictly invariant ($\mathrm{rank} = 2$ for all $\alpha < 1$), where sharing acts under Euclidean gradient descent as an anisotropic optimizer preconditioner with metric tensor condition number scaling as $\kappa \sim (1 - \alpha)^{-1}$.• Algebraic Parameter Decoupling & $C^2$ Routing: Exact orthogonal direct-sum parameter splitting ($\Theta = \Theta_0 \oplus \Theta_1$, $\mathcal{W}_0 \mathcal{W}_1^T = 0$) with guaranteed $C^2$ Quintic Hermite transition routing, eliminating task-gradient conflict on saturated domains ($\cos\rho \equiv 0$).• Cross-Domain Multi-Physics Verification: Rigorously benchmarked across 6 physical regimes, including anisotropic composite heat conduction, 2D magnetostatic inverse design, 3D quadrupole Paul ion traps, viscous Burgers shock dynamics, and 2D Kovasznay Navier-Stokes flows across 8 fixed random seeds. OPEN-SOURCE CODEBASE & REPRODUCIBILITYOfficial Codebase & Benchmark Suite:https://github.com/KartikeyaGangwar/null-space-pinn Permanent Zenodo Archive:https://doi.org/10.5281/zenodo.22132799 All PyTorch implementations, automated verification test suites, benchmark drivers, and raw execution logs across 8 seeds execute directly and are openly available.