Solution Manifold Incompatibility in Multi-Fidelity PINNs: Why Linear Blending Fundamentally Fails for Nonlinear PDEs

Description (v2.3.2): This version presents a refined and expanded analysis of Solution Manifold Incompatibility (SMI) in multi-fidelity physics-informed neural networks for nonlinear partial differential equations. This work identifies a structural limitation of linear multi-fidelity fusion for nonlinear partial differential equations. We introduce Solution Manifold Incompatibility (SMI), a geometric and algebraic obstruction arising from operator non-closure under linear solution blending. Through analysis and experiments on Allen–Cahn, Burgers, and incompressible Navier–Stokes systems, we demonstrate localized, residual-cancelled, and global failure regimes. The results show that residual minimization alone does not certify operator consistency in nonlinear multi-fidelity settings. Relative to earlier versions, this release improves exposition, organization, and clarity, and consolidates the theoretical framework and experimental evidence supporting the central claims. The core mathematical results, diagnostics, and conclusions remain unchanged. This version corresponds to a pre-submission draft intended for archival and reproducibility purposes.