DynGMA: a robust approach for learning stochastic differential equations from data

• URL: http://arxiv.org/abs/2402.14475v2
• Date: Thu, 20 Jun 2024 03:04:10 GMT
• Title: DynGMA: a robust approach for learning stochastic differential equations from data
• Authors: Aiqing Zhu, Qianxiao Li,
• Abstract summary: We introduce novel approximations to the transition density of the parameterized SDE. Our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift diffusion functions. It is capable of handling data with low time resolution and variable, even uncontrollable, time step sizes.
• Score: 13.858051019755283
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method.

Related papers

• Learning Controlled Stochastic Differential Equations [61.82896036131116]
  This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled differential equations with non-uniform diffusion. We provide strong theoretical guarantees, including finite-sample bounds for (L2), (Linfty), and risk metrics, with learning rates adaptive to coefficients' regularity. Our method is available as an open-source Python library.
  arXiv  Detail & Related papers  (2024-11-04T11:09:58Z)
• A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systems [10.820654486318336]
  This study introduces a training-free conditional diffusion model for learning unknown differential equations (SDEs) using data. The proposed approach addresses key challenges in computational efficiency and accuracy for modeling SDEs. The learned models exhibit significant improvements in predicting both short-term and long-term behaviors of unknown systems.
  arXiv  Detail & Related papers  (2024-10-04T03:07:36Z)
• Dynamical Measure Transport and Neural PDE Solvers for Sampling [77.38204731939273]
  We tackle the task of sampling from a probability density as transporting a tractable density function to the target. We employ physics-informed neural networks (PINNs) to approximate the respective partial differential equations (PDEs) solutions. PINNs allow for simulation- and discretization-free optimization and can be trained very efficiently.
  arXiv  Detail & Related papers  (2024-07-10T17:39:50Z)
• On the Trajectory Regularity of ODE-based Diffusion Sampling [79.17334230868693]
  Diffusion-based generative models use differential equations to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models.
  arXiv  Detail & Related papers  (2024-05-18T15:59:41Z)
• Gaussian Mixture Solvers for Diffusion Models [84.83349474361204]
  We introduce a novel class of SDE-based solvers called GMS for diffusion models. Our solver outperforms numerous SDE-based solvers in terms of sample quality in image generation and stroke-based synthesis.
  arXiv  Detail & Related papers  (2023-11-02T02:05:38Z)
• A Geometric Perspective on Diffusion Models [57.27857591493788]
  We inspect the ODE-based sampling of a popular variance-exploding SDE. We establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm.
  arXiv  Detail & Related papers  (2023-05-31T15:33:16Z)
• Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence [65.63201894457404]
  We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of non-linear differential equations. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations.
  arXiv  Detail & Related papers  (2023-05-24T20:43:47Z)
• Rigorous dynamical mean field theory for stochastic gradient descent methods [17.90683687731009]
  We prove closed-form equations for the exact high-dimensionals of a family of first order gradient-based methods. This includes widely used algorithms such as gradient descent (SGD) or Nesterov acceleration.
  arXiv  Detail & Related papers  (2022-10-12T21:10:55Z)
• Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas [3.8325907381729496]
  We propose a data-driven approach to extract governing laws with both (Gaussian) Brownian motion and (non-Gaussian) L'evy motion. We demonstrate that this approach can learn the differential equation with L'evy motion.
  arXiv  Detail & Related papers  (2021-08-28T04:56:51Z)
• Learning effective stochastic differential equations from microscopic simulations: combining stochastic numerics and deep learning [0.46180371154032895]
  We approximate drift and diffusivity functions in effective SDE through neural networks. Our approach does not require long trajectories, works on scattered snapshot data, and is designed to naturally handle different time steps per snapshot.
  arXiv  Detail & Related papers  (2021-06-10T13:00:18Z)
• A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise [5.17900889163564]
  We develop a new data-driven approach to extract governing laws from noisy data sets. First, we establish a feasible theoretical framework, by expressing the drift coefficient, diffusion coefficient and jump measure. We then design a numerical algorithm to compute the drift, diffusion coefficient and jump measure, and thus extract a governing equation with Gaussian and non-Gaussian noise.
  arXiv  Detail & Related papers  (2020-05-07T21:29:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.