FRAME: RETHINKING SPATIO-TEMPORAL FORECASTING FROM OBSERVED BOUNDARIES TO FUTURE FIELDS

Spatio-temporal forecasting asks how an observed relational field should extend into an unknown future. We cast this task as boundary-conditioned extension: history supplies a measured boundary, directed relations define its geometry, and the graph–horizon product domain supports the future field. We introduce FRAME (Future-field Reconstruction through Action-Minimizing Extension), which separates this construction into coordinate-consistent boundary representation, explicit continuation of measured structure, and joint inference of innovation. Robust local affine fibers and sample-conditioned whitened Galerkin projections coordinate the boundary, while a constant-preserving affine trace carries observed level and trajectory forward. Innovation is defined by the unique minimizer of a strongly convex robust action coupling conservative graph exchange, temporal variation, curvature, and boundary attachment, and is approximated by learned solver steps. Our analysis establishes affine factorization, non-expansive projection, uniqueness, and bounded conservative structural influence. Direct numerical verification further shows that every learned update descends the stated action toward its variational target and that four steps closely reproduce the prediction induced by the converged target. Across twelve tasks spanning ten datasets, including ChinaTemp and ChinaWind, FRAME attains the lowest MAE on every task and the lowest RMSE on all six highway and both meteorological benchmarks.