Latent Search Manifolds: Reusing Control Experience for PDE-Constrained Optimization

High-dimensional PDE-constrained control in repeated-task settings faces two distinct online burdens: evaluating candidate controls can require expensive PDE solves, while optimization must still explore a high-dimensional control space. Learned operators can reduce evaluation cost, but a cheap evaluator does not determine where the optimizer should search and can become unreliable when search leaves well-supported regions. We exploit a different reusable asset: the controls produced by previous optimization tasks. Treating these controls as optimization experience, we organize their shared structure into a continuous, low-dimensional search manifold. A sliced-Wasserstein autoencoder provides the representation needed to reconstruct controls while shaping a latent space that can be searched directly. New tasks are then solved by optimizing their objectives on this manifold rather than predicting controls directly. Representation, optimizer, and evaluator can therefore be chosen separately: the manifold determines where to search, derivative-free optimization determines how to search, and either a numerical solver or a frozen neural operator evaluates candidate controls, with final solutions numerically verified. Across eight heterogeneous PDE-control benchmarks, controlled comparisons show that search geometry alone can change solver-verified outcomes by orders of magnitude: on NS2inlets, the objective decreases from 2.325 in the raw control space to 0.0345 on the learned manifold under the same search budget. The learned search space remains compatible with different derivative-free optimizers and with either exact or surrogate evaluation; when PDE solves are expensive, the latter can further reduce online cost. These results identify reusable control geometry as an independent design axis for repeated PDE optimization, complementary to the choices of optimizer and evaluator.