Fig. 2: Visualization of manifold learning, trajectory generation, and generation capabilities of ManGO.

Note that unconditional and conditional samples are generated via ManGO without guidance and with preferred-score guidance, respectively. a, b Manifold and trajectory comparisons for the Branin (SOO) and OmniTest (MOO) tasks. The generated manifold is constructed via ManGO's design-to-score prediction within the feasible region of designs. Close alignment between the ManGO-generated and original manifold, confirming the model’s proficiency in learning complex design-score relationships. Generated trajectories visualize ManGO's score-to-design mapping under minimal score and design constraints, highlighting its capacity to perform targeted denoising toward desired regions. c Branin task: Unconditional samples (green) match preferred scores from the training dataset, while conditional samples (blue) extrapolate beyond the training minimum (grey dashed line). d OmniTest task: Conditional samples better approximate preferred scores and Pareto-dominate the training data (grey) compared to unconditional samples. These results indicate that ManGO effectively reconstructs in-distribution samples during unconditional generation—reflecting well-learned manifold structure—while enabling OOG of superior samples through conditional guidance, demonstrating robust extrapolation based on the learned manifold.