Identifying valuable measurements is one of the main challenges in computational inverse problems, often framed as the optimal experimental design (OED) problem. In this paper, we investigate nonlinear OED within a continuously-indexed design space. This is in contrast to the traditional approaches on selecting experiments from a finite measurement set. This formulation better reflects practical scenarios where measurements are taken continuously across spatial or temporal domains. However, optimizing over a continuously-indexed space introduces computational challenges. To address these, we employ gradient flow and optimal transport techniques, complemented by adaptive strategy for interactive optimization. Numerical results on the Lorenz 63 system and Schr\"odinger equation demonstrate that our solver identifies valuable measurements and achieves improved reconstruction of unknown parameters in inverse problems.

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