Physics-Aligned Finite Maps: Recasting Physics-Constrained Generation as Next-State Prediction

Physics-constrained generative modeling aims to generate realistic physical fields that are both distributionally accurate and physically consistent, with applications to inverse problems, conditional generation, and scientific simulation. Yet these two requirements can impose competing objectives: existing methods typically balance generative and physical losses during training or correct samples at inference, leading to trade-offs among fidelity, consistency, and computational cost. We introduce Physics-Aligned Finite Maps (PAFM), which instead factorizes target construction into distributional progression in free coordinates and physics-based completion. At every step, the sample is advanced toward the data distribution in free physical coordinates, while known physics completes this state into a feasible full field. The two objectives are therefore compatible at each teacher next-state, which serves as both the endpoint of the current transition and the starting point of the next. Physics-constrained generation is thus recast as a sequence of finite next-state predictions with sample-wise compatible teacher targets. We instantiate the prior-to-data sequence with a coarse-to-fine heat bridge, and further distill offline physical completion into a lightweight transition recovery network to suppress rollout drift without physical solves between learned transitions. Across three benchmarks, PAFM achieves low physical residuals, competitive distributional fidelity, and millisecond-scale inference, providing an effective framework for physics-constrained scientific generation.