Shared-Generator Neural Flows for Zero-Support Transition Prediction

Generalizing learned state-transition mechanisms to unobserved condition pairs is a fundamental problem in scientific machine learning, dynamical modeling, and perturbation prediction. Yet learning how a state changes under observed conditions does not determine how to reverse that change or extend it to a larger shift. We study zero-support transition prediction: given a source state and the source and target conditions, the task is to predict the resulting state without examples of the requested transition. Mechanism-Preserving Transport (MPT) ties these predictions together through shared state-dependent vector fields: reversing the condition difference runs the exact flow backward, while doubling it follows the same dynamics for twice as long. To establish when these shared fields also admit a unique interpretation, we characterize the activation patterns that identify their directions up to scaling and permutation, given sufficiently rich grouped short-time observations in fixed coordinates. Our prediction experiments show that the two constraints need not help together. On PDEBench Burgers, MPT reduces direct prediction MSE by 37.8% relative to a zero-preserving conditional ODE over ten seeds. On Shapes3D, keeping reversal but relaxing linear scaling reduces latent MSE by 59.5% relative to MPT. On Norman Perturb-seq, the more flexible zero-preserving model remains more accurate after mean-matching retraining. These controlled comparisons show that preserving an inverse relation can be useful even when extending a change by linear control scaling is not. Reversal and scaling should therefore be chosen separately for unseen-transition prediction, using endpoint accuracy to determine which structure the task supports.