HGAN-SDEs: Learning Neural Stochastic Differential Equations with Hermite-Guided Adversarial Training

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