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Learning Stochastic Dynamics from Data

Ziheng Guo, Igor Cialenco, Ming Zhong

TL;DR

This work develops a noise-guided trajectory-based method to identify drift and diffusion structures from observations of stochastic systems described by ${\rm d}{\bm{x}}_t = {\bm{f}}({\bm{x}}_t)dt + d{\bm{w}}_t$ with state-dependent covariance ${\bm{D}}({\bm{x}})$. By deriving a Girsanov-based likelihood and parameterizing ${\bm{f}}$ in a finite-dimensional basis, the approach learns the drift (and diffusion when possible) from continuous data, with efficient solutions in the diagonal-diffusion case and gradient-based optimization otherwise. It outperforms or complements existing drift-identification methods by explicitly incorporating covariance information, enabling accurate learning under correlated noise and complex diffusion structures. The method is validated on 1D and 2D examples, including polynomial, trigonometric, and neural-network-based drifts, and is shown to produce accurate trajectories and distributional matches as data density varies, offering a principled tool for data-driven stochastic modeling.

Abstract

We present a noise guided trajectory based system identification method for inferring the dynamical structure from observation generated by stochastic differential equations. Our method can handle various kinds of noise, including the case when the the components of the noise is correlated. Our method can also learn both the noise level and drift term together from trajectory. We present various numerical tests for showcasing the superior performance of our learning algorithm.

Learning Stochastic Dynamics from Data

TL;DR

This work develops a noise-guided trajectory-based method to identify drift and diffusion structures from observations of stochastic systems described by with state-dependent covariance . By deriving a Girsanov-based likelihood and parameterizing in a finite-dimensional basis, the approach learns the drift (and diffusion when possible) from continuous data, with efficient solutions in the diagonal-diffusion case and gradient-based optimization otherwise. It outperforms or complements existing drift-identification methods by explicitly incorporating covariance information, enabling accurate learning under correlated noise and complex diffusion structures. The method is validated on 1D and 2D examples, including polynomial, trigonometric, and neural-network-based drifts, and is shown to produce accurate trajectories and distributional matches as data density varies, offering a principled tool for data-driven stochastic modeling.

Abstract

We present a noise guided trajectory based system identification method for inferring the dynamical structure from observation generated by stochastic differential equations. Our method can handle various kinds of noise, including the case when the the components of the noise is correlated. Our method can also learn both the noise level and drift term together from trajectory. We present various numerical tests for showcasing the superior performance of our learning algorithm.
Paper Structure (9 sections, 18 equations, 7 figures, 5 tables)

This paper contains 9 sections, 18 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Learning Framework
  3. Comparison to Others
  4. Examples
  5. Example: sine/cosine drift
  6. Example: Deep Neural Networks
  7. Conclusion
  8. Implementation
  9. Additional Examples

Figures (7)

  • Figure 1: Left: Comparison of ${\bm{f}}$ and $\hat{{\bm{f}}}$. Middle: 5 trajectories generated by ${\bm{f}}$. Right: 5 trajectories generated by $\hat{{\bm{f}}}$ with same noise.
  • Figure 2: Left: Comparison of ${\bm{f}}$ and $\hat{{\bm{f}}}$. Middle: 5 trajectories generated by ${\bm{f}}$. Right: 5 trajectories generated by $\hat{{\bm{f}}}$ with same noise.
  • Figure 3: One-dimensional Polynomial Drift Comparison Summary
  • Figure 4: Two-dimensional Polynomial Trajectory Comparison
  • Figure 5: Two-dimensional Polynomial Comparison of ${\bm{f}}$ and $\hat{{\bm{f}}}$. (a) Surface of Dimension 1 (b) Surface of Dimension 2
  • ...and 2 more figures