Beyond $L_2$: Sobolev-Kinematic Flow Matching for Extreme Sparse Physical Inversion

Physical inversion from extremely sparse sensors demands expressive priors. Prevailing paradigms either rely on $L_2$ metrics—causing high-frequency artifacts—or depend on specific differentiable partial differential equations (PDEs) solvers at inference, leading to severe collapse under observation sparsity, massive computational overhead and poor cross-scenario generalization. To transcend this, we propose Sobolev-Kinematic Flow Matching (SK-FM). First, breaking away from standard $L_2$ paradigms, we embed the joint probability flow for scalar fields and spatial gradients directly within the $W^{1,2}$ Sobolev space, implicitly learning differential operators without heuristic interpolations. Second, we introduce an equation-free Inference Kinematic Guidance. Leveraging ODE-based terminal state estimation to formulate geometric objectives—spatial derivative consistency and a curl-free constraint ($\nabla \times \mathbf{g}\equiv 0$)—we steer the trajectory toward the valid irrotational manifold. This propagates kinematic corrections back into the Sobolev space, successfully bypassing explicit numerical solvers. Extensive experiments across Darcy flow, Helmholtz, Structural Health Monitoring, and Navier-Stokes demonstrate that SK-FM achieves state-of-the-art inversion fidelity at $128\times 128$ resolution using merely 128 random observations (<0.79%). Remarkably, SK-FM reduces average inversion error by 78% compared to baselines. Finally, uncertainty quantification highlights its potential for real-world engineering applications.