A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systems

• URL: http://arxiv.org/abs/2412.03970v1
• Date: Thu, 05 Dec 2024 08:38:30 GMT
• Title: A Data-Driven Framework for Discovering Fractional Differential Equations in Complex Systems
• Authors: Xiangnan Yu, Hao Xu, Zhiping Mao, HongGuang Sun, Yong Zhang, Dongxiao Zhang, Yuntian Chen,
• Abstract summary: This study introduces a stepwise data-driven framework for discovering fractional differential equations (FDEs) directly from data. Our framework applies deep neural networks as surrogate models for denoising and reconstructing sparse and noisy observations. We validate the framework across various datasets, including synthetic anomalous diffusion data and experimental data on the creep behavior of frozen soils.
• Score: 8.206685537936078
• License:
• Abstract: In complex physical systems, conventional differential equations often fall short in capturing non-local and memory effects, as they are limited to local dynamics and integer-order interactions. This study introduces a stepwise data-driven framework for discovering fractional differential equations (FDEs) directly from data. FDEs, known for their capacity to model non-local dynamics with fewer parameters than integer-order derivatives, can represent complex systems with long-range interactions. Our framework applies deep neural networks as surrogate models for denoising and reconstructing sparse and noisy observations while using Gaussian-Jacobi quadrature to handle the challenges posed by singularities in fractional derivatives. To optimize both the sparse coefficients and fractional order, we employ an alternating optimization approach that combines sparse regression with global optimization techniques. We validate the framework across various datasets, including synthetic anomalous diffusion data, experimental data on the creep behavior of frozen soils, and single-particle trajectories modeled by L\'{e}vy motion. Results demonstrate the framework's robustness in identifying the structure of FDEs across diverse noise levels and its capacity to capture integer-order dynamics, offering a flexible approach for modeling memory effects in complex systems.

Related papers

• Deep Generative Modeling for Identification of Noisy, Non-Stationary Dynamical Systems [3.1484174280822845]
  We focus on finding parsimonious ordinary differential equation (ODE) models for nonlinear, noisy, and non-autonomous dynamical systems. Our method, dynamic SINDy, combines variational inference with SINDy (sparse identification of nonlinear dynamics) to model time-varying coefficients of sparse ODEs.
  arXiv  Detail & Related papers  (2024-10-02T23:00:00Z)
• Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator [0.0]
  Closure models are widely used in simulating complex multiscale dynamical systems such as turbulence and the earth system. For systems without a clear scale, generalization deterministic and local closure models often lack enough capability. We propose a datadriven modeling framework for constructing neural operator and non-local closure models.
  arXiv  Detail & Related papers  (2024-08-06T05:21:31Z)
• 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)
• Automating the Discovery of Partial Differential Equations in Dynamical Systems [0.0]
  We present an extension to the ARGOS framework, ARGOS-RAL, which leverages sparse regression with the recurrent adaptive lasso to identify PDEs automatically. We rigorously evaluate the performance of ARGOS-RAL in identifying canonical PDEs under various noise levels and sample sizes. Our results show that ARGOS-RAL effectively and reliably identifies the underlying PDEs from data, outperforming the sequential threshold ridge regression method in most cases.
  arXiv  Detail & Related papers  (2024-04-25T09:23:03Z)
• 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)
• 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. This work introduces a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. 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)
• A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization [1.2882319878552302]
  This paper develops a new iterative learning algorithm for complex turbulent systems with partial observations. It alternates between identifying model structures, recovering unobserved variables, and estimating parameters. Numerical experiments show that the new algorithm succeeds in identifying the model structure and providing suitable parameterizations for many complex nonlinear systems.
  arXiv  Detail & Related papers  (2022-08-19T00:35:03Z)
• 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)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)
• 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.