Nansde-net: A neural sde framework for generating time series with memory

Paper: arXiv 2602.08182

Authors: Hiromu Ozai, Kei Nakagawa

Abstract

Modeling time series with long- or short-memory characteristics is a fundamental challenge in many scientific and engineering domains. While fractional Brownian motion has been widely used as a noise source to capture such memory effects, its incompatibility with Itô calculus limits its applicability in neural stochastic differential equation~(SDE) frameworks. In this paper, we propose a novel class of noise, termed Neural Network-kernel ARMA-type noise~(NA-noise), which is an Itô-process-based alternative capable of capturing both long- and short-memory behaviors. The kernel function defining the noise structure is parameterized via neural networks and decomposed into a product form to preserve the Markov property. Based on this noise process, we develop NANSDE-Net, a generative model that extends Neural SDEs by incorporating NA-noise. We prove the theoretical existence and uniqueness of the solution under mild conditions and derive an efficient backpropagation scheme for training. Empirical results on both synthetic and real-world datasets demonstrate that NANSDE-Net matches or outperforms existing models, including fractional SDE-Net, in reproducing long- and short-memory features of the data, while maintaining computational tractability within the Itô calculus framework.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a strong theoretical bridge between Neural SDEs and non-Markovian memory structures by parameterizing the ARMA-type Volterra kernel with neural networks inside the Itô calculus framework. While the mathematical foundation and proof of existence/uniqueness are rigorous, the empirical validation relies on basic simulated/real time-series settings and would benefit from more extensive financial backtests.

Research Flowchart

  flowchart TD
    A[Research Goal: Model time series with long/short memory in Neural SDEs] --> B[Input Data: Synthetic & Real-world Multi-Asset Time Series]
    B --> C[Propose NA-Noise: Itô-based ARMA noise with neural network kernels]
    C --> D[Develop NANSDE-Net: Integrate NA-noise into Neural SDE framework]
    D --> E[Derive Efficient Backpropagation & Prove Solution Uniqueness]
    E --> F[Outcomes: Outperforms fSDE-Net in memory capture while keeping Itô tractability]

Cited References (16)

1. Lf-Net:Generating Fractional Time-Series with Latent Fractional-Net (2024) — link
2. Neural Stochastic Differential Equations with Change Points: A Generative Adversarial Approach (2023) — link
3. Constrained Dynamical Neural ODE for Time Series Modelling: A Case Study on Continuous Emotion Prediction (2023) — link
4. ItôWave: Itô Stochastic Differential Equation is all You Need for Wave Generation (2022) — link
5. Fractional SDE-Net: Generation of Time Series Data with Long-term Memory (2022) — link
6. Deep Neural Networks with Flexible Complexity While Training Based on Neural Ordinary Differential Equations (2021) — link
7. Neural SDEs as Infinite-Dimensional GANs (2021) — link
8. SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimates (2020) — link

0 of 16 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.