Conditional flow matching can transfer too: a correction on top of a frozen source representation cuts energy discrepancy 26.4% on five-dimensional responses

Transfer for conditional flow matching splits into two uses: freeze the source representation, or learn a target correction on top of it. Earlier shared-embedding transfer fixed the representation learned from source data, so the target fit only learned a velocity head and any source-target representation mismatch stayed in the target risk. In simulations with related five-dimensional representations and 10000 source samples, unpenalized TFM-Adapt reached an energy discrepancy of 0.2325 against 0.3158 for the frozen-representation TFM-Freeze, 26.4% lower.

The correction does not relearn the whole representation; it adds a smooth residual built from a few context coordinates to the source encoder. TFM-Freeze trains only the target velocity head, while TFM-Adapt jointly trains the target head and a residual encoder, with a functional anchoring penalty pulling the new representation toward the source fit. The simulations cover nonlinear sufficient representations, heteroscedastic responses and a two-component mixture response, with target sample size fixed and source sample size varied over a grid, using 100 independent training repetitions per setting.

Target validation returns the generator to a target-only version when transfer does not help. Without validation, adaptive transfer produced harmful repetitions in settings where source and target representations are unrelated; with validation, each setting had at most one harmful repetition. At 10000 source samples in the unrelated-representation setting, both adaptive procedures recovered target-only flow matching exactly in all three dimensions, with no harmful repetition in 100.

The demonstrated scope is frozen transfer and functionally anchored correction for conditional flow matching, with simulations covering nonlinear representations, heteroscedasticity and mixture responses, and real-data applications to superconducting critical temperature and diamond prices; the validation rule addresses a finite dictionary of candidate generators and assumes bounded endpoint support.

The guarantees rest on smoothness, flow-regularity and correctability assumptions, and the constants are not claimed to be uniform in growing ambient dimension; when source and target representations are unrelated, validation can also reject useful transfer, and for related scalar responses at 1000 source samples the positive- and zero-penalty fits were retained in only 49 and 51 repetitions respectively.