Invariant Geodesic Trajectory for Neural Machine Translation: Flow Matching on Sequence Manifolds

Conventional neural machine translation (NMT) models formulate translation as a mapping between independent, discrete tokens in flat Euclidean space, failing to capture the continuous structural and semantic trajectories of sentences. While semantic graph augmented models address this by injecting structured priors, they rely on noisy, computationally expensive external parsers. In this work, we introduce the Topological Geometric Flow Invariant framework, realized as Geodesic Flow Matching on Sequence Manifolds (GeFLoSM), a novel paradigm that reformulates NMT as continuous boundary value mechanics on a Poincaré ball manifold. GeFLoSM represents sentences as continuous, piecewise geodesic splines on the manifold. We parameterize the sequential order of source tokens as a geodesic spline, which is integrated along its path via a neural ODE to yield a length invariant trajectory representation Ψ that dynamically conditions a tangent space Transformer decoder. To align source and target geometries, we train a Riemannian conditional flow matching (CFM) velocity field that transports source splines to target splines across flow time. Crucially, we identify a severe train/inference mismatch that reduces validation BLEU to 0.60 when auxiliary geometric logits are naively fused during generation, and present decode decoupling as the structural solution enabling joint end to end training with stable inference. Extensive evaluations across five benchmarks (WMT16 EN DE, IWSLT15 EN VI, WMT14 EN DE, WMT16 EN RO, and IWSLT14 DE EN) demonstrate that GeFLoSM establishes state of the art translation performance, consistently out performing both standard Euclidean Transformers and structured semantic graph augmented baselines. Specifically, GeFLoSM achieves up to +3.1 BLEU over the vanilla Transformer and +1.3 BLEU over prior state of the art AMR augmented Transformers on WMT16 EN DE(NCv11).