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

TL;DR

HGAN-SDEs introduces a Hermite-guided adversarial training framework for neural SDEs, improving efficiency and stability in modeling complex stochastic processes with superior sample quality.

Contribution

The paper proposes HGAN-SDEs, a novel GAN-based method using Hermite functions for efficient and stable learning of neural SDEs, addressing limitations of previous discriminator architectures.

Findings

Achieves better sample quality than existing models.

Reduces computational complexity in training.

Demonstrates improved stability and convergence.

Abstract

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