GalerkinFlow: Flow Matching in Function Space via Galerkin Projection

Existing time-series generative models evolve every sampled value, so the generative state grows with window length even when the underlying signal is compact. Functional approaches lift this to infinite-dimensional spaces but still discretize at inference, leaving the state tied to the sampling grid. We introduce GalerkinFlow, which sidesteps both limitations by shifting generation entirely into a low-dimensional spline coefficient space. Each observed window is projected onto a fixed cubic B-spline basis via ridge fitting, a Transformer learns the joint $K\times D$ coefficient distribution through standard flow matching, and a fixed linear synthesis map decodes samples on any prescribed grid. Because synthesis is an isometry in the spline Gram metric, the squared functional Wasserstein error decomposes exactly into a projection floor set by the basis and a coefficient-transport term that training can reduce — providing an explicit error budget absent from prior work. A further bound accounts for fitting coefficients from finite, possibly incomplete, observations. GalerkinFlow achieves the lowest Context-FID on all six regular benchmarks, outperforms \smf on irregular-to-irregular generation at every tested nonzero drop rate, and trains 2.5–4.6$\times$ faster across the regular benchmarks.