HGAN-SDEs: Learning Neural Stochastic Differential Equations with Hermite-Guided Adversarial Training
This work addresses a computational bottleneck in modeling continuous-time stochastic processes for fields like physics and finance, representing an incremental improvement over prior methods.
The paper tackles the problem of designing an efficient and stable discriminator for learning Neural Stochastic Differential Equations (SDEs) using Generative Adversarial Networks (GANs), and introduces HGAN-SDEs, which leverages Neural Hermite functions to achieve superior sample quality and learning efficiency compared to existing models.
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