Learning Stochastic Dynamical Systems with Structured Noise

• URL: http://arxiv.org/abs/2503.01077v1
• Date: Mon, 03 Mar 2025 00:40:53 GMT
• Title: Learning Stochastic Dynamical Systems with Structured Noise
• Authors: Ziheng Guo, James Greene, Ming Zhong,
• Abstract summary: Due to the availability of large-scale data sets, there is growing interest in learning models from observations with noise.<n>We present a nonparametric framework to learn both the drift and diffusion terms in systems of SDEs where the noise is singular.<n>We provide an algorithm for constructing estimators given trajectory data and demonstrate the effectiveness of our methods.
• Score: 12.056775765064266
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest in learning mechanistic models from observations with stochastic noise. In this work, we present a nonparametric framework to learn both the drift and diffusion terms in systems of SDEs where the stochastic noise is singular. Specifically, inspired by second-order equations from classical physics, we consider systems which possess structured noise, i.e. noise with a singular covariance matrix. We provide an algorithm for constructing estimators given trajectory data and demonstrate the effectiveness of our methods via a number of examples from physics and biology. As the developed framework is most naturally applicable to systems possessing a high degree of dimensionality reduction (i.e. symmetry), we also apply it to the high dimensional Cucker-Smale flocking model studied in collective dynamics and show that it is able to accurately infer the low dimensional interaction kernel from particle data.

Related papers

• A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systems [8.206685537936078]
  This study introduces a stepwise data-driven framework for discovering fractional differential equations (FDEs) directly from data.<n>Our framework applies deep neural networks as surrogate models for denoising and reconstructing sparse and noisy observations.<n>We validate the framework across various datasets, including synthetic anomalous diffusion data and experimental data on the creep behavior of frozen soils.
  arXiv  Detail & Related papers  (2024-12-05T08:38:30Z)
• 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)
• Zero-shot Imputation with Foundation Inference Models for Dynamical Systems [5.549794481031468]
  We offer a fresh perspective on the classical problem of imputing missing time series data, whose underlying dynamics are assumed to be determined by ODEs.<n>We propose a novel supervised learning framework for zero-shot time series imputation, through parametric functions satisfying some (hidden) ODEs.<n>We empirically demonstrate that one and the same (pretrained) recognition model can perform zero-shot imputation across 63 distinct time series with missing values.
  arXiv  Detail & Related papers  (2024-02-12T11:48:54Z)
• Equivariant Graph Neural Operator for Modeling 3D Dynamics [148.98826858078556]
  We propose Equivariant Graph Neural Operator (EGNO) to directly models dynamics as trajectories instead of just next-step prediction. EGNO explicitly learns the temporal evolution of 3D dynamics where we formulate the dynamics as a function over time and learn neural operators to approximate it. Comprehensive experiments in multiple domains, including particle simulations, human motion capture, and molecular dynamics, demonstrate the significantly superior performance of EGNO against existing methods.
  arXiv  Detail & Related papers  (2024-01-19T21:50:32Z)
• HyperSINDy: Deep Generative Modeling of Nonlinear Stochastic Governing Equations [5.279268784803583]
  We introduce HyperSINDy, a framework for modeling dynamics via a deep generative model of sparse governing equations from data. Once trained, HyperSINDy generates dynamics via a differential equation whose coefficients are driven by a white noise. In experiments, HyperSINDy recovers ground truth governing equations, with learnedity scaling to match that of the data.
  arXiv  Detail & Related papers  (2023-10-07T14:41:59Z)
• Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference [9.599029891108229]
  We focus on an operator inference methodology that builds dynamical models. For inference, we aim to learn the operators of a model by setting up an appropriate optimization problem. We present several numerical examples, illustrating the preservation of stability and discussing its comparison with the existing state-of-the-art approach to infer operators.
  arXiv  Detail & Related papers  (2023-08-26T09:00:31Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• Score-based Diffusion Models in Function Space [137.70916238028306]
  Diffusion models have recently emerged as a powerful framework for generative modeling.<n>This work introduces a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space.<n>We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Capturing Actionable Dynamics with Structured Latent Ordinary Differential Equations [68.62843292346813]
  We propose a structured latent ODE model that captures system input variations within its latent representation. Building on a static variable specification, our model learns factors of variation for each input to the system, thus separating the effects of the system inputs in the latent space.
  arXiv  Detail & Related papers  (2022-02-25T20:00:56Z)
• Using scientific machine learning for experimental bifurcation analysis of dynamic systems [2.204918347869259]
  This study focuses on training universal differential equation (UDE) models for physical nonlinear dynamical systems with limit cycles. We consider examples where training data is generated by numerical simulations, whereas we also employ the proposed modelling concept to physical experiments. We use both neural networks and Gaussian processes as universal approximators alongside the mechanistic models to give a critical assessment of the accuracy and robustness of the UDE modelling approach.
  arXiv  Detail & Related papers  (2021-10-22T15:43:03Z)
• Using Data Assimilation to Train a Hybrid Forecast System that Combines Machine-Learning and Knowledge-Based Components [52.77024349608834]
  We consider the problem of data-assisted forecasting of chaotic dynamical systems when the available data is noisy partial measurements. We show that by using partial measurements of the state of the dynamical system, we can train a machine learning model to improve predictions made by an imperfect knowledge-based model.
  arXiv  Detail & Related papers  (2021-02-15T19:56:48Z)

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.