Beyond Parameter-Space Conflict: Intrinsic Function-Space Geometry for Physics-Constrained Flow Matching

Physics-constrained generative modeling requires balancing distribution fidelity with physical consistency, two objectives that can induce conflicting optimization directions. Existing approaches typically resolve this conflict using parameter gradients, but we show that the resulting geometry depends on the chosen parameterization: function-preserving reparameterizations can substantially alter measured conflict without changing the represented model. Because multi-objective rules use this geometry to determine the update itself, equivalent parameterizations can induce different optimization decisions for the same model function. Based on this observation, we introduce Intrinsic Geometry Flow Matching (IGFM), which defines objective conflict over realizable first-order changes in function space and constructs a common-descent direction within this intrinsic geometry. For a fixed observation map and output metric, the resulting ideal direction is invariant to locally invertible changes of parameter coordinates and admits a scale-free common-descent guarantee. Across Darcy flow, Kolmogorov flow, and Structural Topology Optimization, IGFM reduces the primary physics or engineering error by $40.0\%$, $22.4\%$, and $55.7\%$, respectively, relative to the strongest PBFM baseline, while maintaining comparable task-specific fidelity. These theoretical and empirical findings support realizable function-space geometry as a principled framework for resolving physics–distribution conflicts in generative modeling.