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

Yuanjian Xu, Yuan Shuai, Jianing Hao, Guang Zhang

TL;DR

HGAN-SDEs is introduced, a novel GAN-based framework that leverages Neural Hermite functions to construct a structured and efficient discriminator and establishes the universal approximation property of the framework for a broad class of SDE-driven distributions and theoretically characterize its convergence behavior.

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

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

TL;DR

HGAN-SDEs is introduced, a novel GAN-based framework that leverages Neural Hermite functions to construct a structured and efficient discriminator and establishes the universal approximation property of the framework for a broad class of SDE-driven distributions and theoretically characterize its convergence behavior.

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
Paper Structure (11 sections, 3 theorems, 6 equations, 2 figures, 4 tables)

This paper contains 11 sections, 3 theorems, 6 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Methodology
  4. Neural SDEs as the GAN Generators
  5. Neural Hermite Functions as GAN Discriminators
  6. Expressivity of Hermite-Based Discriminator
  7. Experiments
  8. Experiment Settings
  9. Experimental Results Analysis
  10. More Explorations
  11. Conclusion

Key Result

Theorem 1

The Hermite functions $\{\psi_n(x)\}_{n=0}^\infty$ satisfy the orthonormality condition: $\int_{\mathbb{R}} \psi_n(x) \psi_m(x) dx = \delta_{nm},$ where $\delta_{nm}$ is the Kronecker delta.

Figures (2)

  • Figure 1: Comparison between HGAN-SDEs and Autoformer on the CIR dataset. The figure shows how predicted sequence distributions evolve across time points, revealing that HGAN-SDEs maintain smoother and more consistent dynamics, while Autoformer tends to generate more dispersed outputs.
  • Figure 2: The figure shows the changes in MISE with the increase of neural Hermite terms in HGAN-SDEs. We present the patterns on three datasets, with the parameter configurations consistent with the previous experiments.

Theorems & Definitions (5)

  • Theorem 1: Orthonormality of Hermite functions abramowitz1972handbookeijndhoven1990new
  • Theorem 2: Convergence of Hermite Expansion
  • proof : Proof Sketch
  • Theorem 3: Discriminative Power of Hermite Span
  • proof