A Parallel Adaptive Localized Kriging‑Believer Algorithm for Optimizing Fish-School Propulsion Performance

Fish schooling problems involve various flow physics like vortex interactions, quasi steady forces and added mass effects, each comes with different length and amplitude scales. Therefore, optimization in fish schooling formation problems is particularly challenging due to the combination of the multi-modal, multi-scale search space and the computing cost of using High-fidelity Computational Fluid Dynamics (CFD) simulations as the sampling tool. This combination often leads to either inefficient sampling caused by Gaussian process prior mismatch or prohibitive costs in complex surrogate model training. Exploiting the deterministic nature of CFD evaluations, we developed a parallel optimization method that adaptively shrinks the search space around promising regions. By increasing sampling density locally, the Kriging Believer is gradually enforced to exactly interpolate through the true landscape within these regions, thereby effectively mitigating the inefficient sampling caused by the Gaussian process prior mismatch. As a result, the method delivers rapid and reliable convergence performance within short time frames. We further demonstrate the convergence performance of this parallel optimization method in a couple of canonical fish schooling formation problems. The proposed method converged to formations with promising propulsive performance with 95~99% fewer CFD calls than grid search. The proposed method is also able to largely maintain its sampling efficiency when parallelized to a sampling size of around 40 and can therefore potentially alleviate the parallel scalability bottleneck of the CFD solver.