Neural Stochastic Differential Equations (Neural SDEs) provide a principled framework for modeling continuous-time stochastic processes and have been widely adopted in fields ranging from physics to finance. Recent advances suggest that Generative Adversarial Networks (GANs) offer a promising solution to learning the complex path distributions induced by SDEs. However, a critical bottleneck lies in designing a discriminator that faithfully captures temporal dependencies while remaining computationally efficient. Prior works have explored Neural Controlled Differential Equations (CDEs) as discriminators due to their ability to model continuous-time dynamics, but such architectures suffer from high computational costs and exacerbate the instability of adversarial training. To address these limitations, we introduce HGAN-SDEs, a novel GAN-based framework that leverages Neural Hermite functions to construct a structured and efficient discriminator. Hermite functions provide an expressive yet lightweight basis for approximating path-level dynamics, enabling both reduced runtime complexity and improved training stability. We establish the universal approximation property of our framework for a broad class of SDE-driven distributions and theoretically characterize its convergence behavior. Extensive empirical evaluations on synthetic and real-world systems demonstrate that HGAN-SDEs achieve superior sample quality and learning efficiency compared to existing generative models for SDEs

翻译：神经随机微分方程（Neural SDEs）为建模连续时间随机过程提供了一个原则性框架，并已广泛应用于从物理学到金融学等多个领域。最新研究表明，生成对抗网络（GANs）为学习SDEs所诱导的复杂路径分布提供了一种有前景的解决方案。然而，关键瓶颈在于设计一个既能忠实捕捉时间依赖性、又能保持计算效率的判别器。先前研究探索了使用神经受控微分方程（CDEs）作为判别器，因其具备建模连续时间动态的能力，但此类架构存在计算成本高昂的问题，并加剧了对抗训练的不稳定性。为应对这些局限性，我们提出了HGAN-SDEs——一种基于GAN的新型框架，其利用神经Hermite函数构建结构化的高效判别器。Hermite函数为逼近路径级动态提供了表达力强且轻量化的基函数，从而同时降低了运行时复杂度并提升了训练稳定性。我们证明了该框架对一大类SDE驱动分布具有通用逼近性质，并从理论上刻画了其收敛行为。在合成系统与真实系统上的大量实证评估表明，相较于现有SDE生成模型，HGAN-SDEs在样本质量和学习效率方面均实现了更优的性能。