Neural SDEs as Infinite-Dimensional GANs (2102.03657v2)

Abstract: Stochastic differential equations (SDEs) are a staple of mathematical modelling of temporal dynamics. However, a fundamental limitation has been that such models have typically been relatively inflexible, which recent work introducing Neural SDEs has sought to solve. Here, we show that the current classical approach to fitting SDEs may be approached as a special case of (Wasserstein) GANs, and in doing so the neural and classical regimes may be brought together. The input noise is Brownian motion, the output samples are time-evolving paths produced by a numerical solver, and by parameterising a discriminator as a Neural Controlled Differential Equation (CDE), we obtain Neural SDEs as (in modern machine learning parlance) continuous-time generative time series models. Unlike previous work on this problem, this is a direct extension of the classical approach without reference to either prespecified statistics or density functions. Arbitrary drift and diffusions are admissible, so as the Wasserstein loss has a unique global minima, in the infinite data limit any SDE may be learnt. Example code has been made available as part of the \texttt{torchsde} repository.

• The paper introduces a framework that redefines neural SDEs as GAN generators by leveraging neural CDE discriminators to transform Brownian motion into continuous-time paths.
• The paper demonstrates enhanced performance in modeling dynamic processes, outperforming methods like latent ODEs and continuous-time flow processes through applications in finance and environmental data.
• The paper advances stochastic modeling by parameterizing drift and diffusion through neural networks, allowing for learned infinite-dimensional statistics without predefining density functions.

Insightful Overview of "Neural SDEs as Infinite-Dimensional GANs"

The paper "Neural SDEs as Infinite-Dimensional GANs" by Patrick Kidger et al. proposes a novel framework for conceptualizing neural stochastic differential equations (SDEs) within the paradigm of generative adversarial networks (GANs). Neural SDEs are cast as continuous-time generative models capable of modeling complex, time-evolving phenomena, providing a bridge between classical mathematical approaches and modern machine learning techniques.

Core Contributions

The authors unveil a methodology that reimagines neural SDEs as GANs operating in infinite-dimensional spaces. This framework is based on treating SDEs as transformations of Brownian motion to produce time-evolving sample paths. The authors utilize a neural controlled differential equation (CDE) as a discriminator, thus integrating SDEs into the GAN framework. This approach enables a direct extension of the classical SDE model-fitting approach, obviating the need for prespecified statistics or density functions and instead utilizing learned statistics.

Across four key applications—synthetic time-dependent Ornstein–Uhlenbeck processes, Google/Alphabet stock prices, air quality in Beijing, and SGD-trained weights—the framework demonstrates enhanced modeling capabilities. The framework shows promising results in capturing the dynamics of these processes, outperforming alternate models like latent ODEs and continuous-time flow processes (CTFPs) in various metrics.

Methodology and Results

The central innovation lies in conceptualizing the SDE model as a generator within a GAN framework, with the neural CDE functioning as the discriminator. This is achieved by parametrizing the vector fields of the SDE using neural networks. The generator transforms Brownian noise into a continuous-time path, while the discriminator uses the Wasserstein metric to compare real and generated data. This scheme allows for the learning of arbitrary drift and diffusion, theoretically enabling the model to approximate any target SDE in the infinite data limit.

The authors underscore the advantages of this approach through empirical results. For instance, in modeling stock price dynamics, the neural SDE reported superior classification accuracy, prediction, and maximum mean discrepancy (MMD) compared to the competitors. Similarly, in the air quality dataset, the neural SDE showed strengths in prediction and MMD, indicating its robustness in capturing real-world stochastic processes.

Implications and Future Directions

The integration of neural SDEs and GANs presents significant implications for stochastic modeling and time-series analysis. This framework not only enhances the flexibility and generalization capabilities of SDEs but also opens pathways for integrating deep learning and differential equations more seamlessly. Practically, this can lead to advanced modeling techniques in finance, biology, and beyond, where understanding temporal stochastic processes is crucial.

Theoretically, this work prompts further investigation into the convergence properties and computational methodologies of neural SDEs. Future work could explore optimizing training regimes and exploring richer network architectures to further leverage the neural SDE framework's capacity for modeling complex stochastic dependencies.

In conclusion, the paper outlines a thoughtful advancement in time-series modeling, offering a robust generative model that leverages the strengths of both stochastic differential equations and generative adversarial networks. This synthesis could form the bedrock of many future developments in predictive modeling and simulation in AI and allied fields.