We study Neural Optimal Transport in infinite-dimensional Hilbert spaces. In non-regular settings, Semi-dual Neural OT often generates spurious solutions that fail to accurately capture target distributions. We analytically characterize this spurious solution problem using the framework of regular measures, which generalize Lebesgue absolute continuity in finite dimensions. To resolve ill-posedness, we extend the semi-dual framework via a Gaussian smoothing strategy based on Brownian motion. Our primary theoretical contribution proves that under a regular source measure, the formulation is well-posed and recovers a unique Monge map. Furthermore, we establish a sharp characterization for the regularity of smoothed measures, proving that the success of smoothing depends strictly on the kernel of the covariance operator. Empirical results on synthetic functional data and time-series datasets demonstrate that our approach effectively suppresses spurious solutions and outperforms existing baselines.

翻译：我们研究了无限维希尔伯特空间中的神经最优传输问题。在非正则设定下，半对偶神经最优传输常产生虚假解，无法准确捕捉目标分布。我们利用正则测度框架对此虚假解问题进行解析表征，该框架推广了有限维中的勒贝格绝对连续性概念。为解决不适定性问题，我们基于布朗运动的高斯平滑策略扩展了半对偶框架。主要理论贡献在于证明：在正则源测度下，该公式是适定的且能恢复唯一的蒙日映射。此外，我们建立了平滑测度正则性的精确表征，证明平滑的成功严格依赖于协方差算子的核。在合成函数数据与时间序列数据集上的实验结果表明，本方法能有效抑制虚假解并超越现有基线。