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.

翻译：在计算反问题中，识别有价值的测量是主要挑战之一，通常被表述为最优实验设计问题。本文研究连续索引设计空间中的非线性最优实验设计，这与传统从有限测量集中选择实验的方法形成对比。该公式更好地反映了在空间或时间域上连续进行测量的实际场景。然而，在连续索引空间上进行优化带来了计算挑战。为解决这些问题，我们采用梯度流和最优传输技术，并辅以自适应策略进行交互式优化。在Lorenz 63系统和薛定谔方程上的数值结果表明，我们的求解器能够识别有价值的测量，并在反问题中实现对未知参数的改进重建。