Physics-Embedded Flow Matching: Governing the Local Vector Field with a Known Physiological Relation

Physiological signal translation often suffers from limited paired data, motivating the use of governing relationships between source and target measurements as additional supervision. Existing physics-informed models constrain supervised predictions, while their flow matching counterparts constrain reconstructed states, generated endpoints, or generic properties of the transport field. We introduce a physics-informed flow matching framework that instead imposes the physiological relationship directly on the learned local velocity field through a physical constraint. We derive a closed-form velocity-space residual that softly enforces physical consistency at each flow time, and we present a unified error analysis characterizing how physics regularization and multi-step integration can reduce prediction error. Using a learned two-element Windkessel model, we evaluate translation from virtual aortic flow and from measured bioimpedance to pressure against supervised and flow matching baselines with and without physics constraints. Our Physics Embedded Flow Matching (PEFM) method achieves the lowest waveform RMSE on both datasets. On measured data, it achieves $11.8\%$ lower RMSE than the baseline closest in physical consistency and $63.6\%$ lower physics residual than the baseline closest in RMSE. These results support the use of governing relationships to jointly improve predictive accuracy and physical consistency.