Physics-Informed One-Step Generation on Long-Short Flow Maps

We propose Physics-Informed Long-Short Flow Maps with Direct Guidance (PI-LSFM), a one-step generative method for producing data-aligned and physically valid samples. Existing one-step physics-informed generative models must jointly control data fidelity and physical constraints without the iterative correction of multi-step samplers, making it difficult to coordinate indirect surrogate supervision with strong physics enforcement. PI-LSFM addresses this challenge by decomposing generation into a long flow map for global transport and a near-terminal short flow map, where closed-form data and physics potentials provide directly computable guidance targets instead of indirect regression surrogates. The short map guides the long map during training, yielding a single one-step generator that requires neither staged distillation nor inference-time correction. Across Darcy flow, Kolmogorov flow, dynamic stall, and topology optimization, PI-LSFM achieves a favorable trade-off among distributional fidelity, physical residuals, and inference efficiency while using only one function evaluation at inference.