Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systems

• URL: http://arxiv.org/abs/2410.10023v1
• Date: Sun, 13 Oct 2024 21:48:51 GMT
• Title: Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systems
• Authors: Ashish Pal, Sutanu Bhowmick, Satish Nagarajaiah,
• Abstract summary: The proposed method is demonstrated to discover various differential equations at various noise levels, including three-dimensional, fourth-order, and stiff equations. The parameter estimation converges accurately to the true values with a small coefficient of variation, suggesting robustness to the noise.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Sparse system identification of nonlinear dynamic systems is still challenging, especially for stiff and high-order differential equations for noisy measurement data. The use of highly correlated functions makes distinguishing between true and false functions difficult, which limits the choice of functions. In this study, an equation discovery method has been proposed to tackle these problems. The key elements include a) use of B-splines for data fitting to get analytical derivatives superior to numerical derivatives, b) sequentially regularized derivatives for denoising (SRDD) algorithm, highly effective in removing noise from signal without system information loss, c) uncorrelated component analysis (UCA) algorithm that identifies and eliminates highly correlated functions while retaining the true functions, and d) physics-informed spline fitting (PISF) where the spline fitting is updated gradually while satisfying the governing equation with a dictionary of candidate functions to converge to the correct equation sequentially. The complete framework is built on a unified deep-learning architecture that eases the optimization process. The proposed method is demonstrated to discover various differential equations at various noise levels, including three-dimensional, fourth-order, and stiff equations. The parameter estimation converges accurately to the true values with a small coefficient of variation, suggesting robustness to the noise.

Related papers

• Accelerated zero-order SGD under high-order smoothness and overparameterized regime [79.85163929026146]
  We present a novel gradient-free algorithm to solve convex optimization problems. Such problems are encountered in medicine, physics, and machine learning. We provide convergence guarantees for the proposed algorithm under both types of noise.
  arXiv  Detail & Related papers  (2024-11-21T10:26:17Z)
• 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)
• Weak Collocation Regression for Inferring Stochastic Dynamics with L\'{e}vy Noise [8.15076267771005]
  We propose a weak form of the Fokker-Planck (FP) equation for extracting dynamics with L'evy noise. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems.
  arXiv  Detail & Related papers  (2024-03-13T06:54:38Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce 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)
• WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming [5.027714423258538]
  We propose a general and robust framework to recover differential equations using a weak formulation. For each sparsity level, Subspace Pursuit is utilized to find an initial set of support from the large dictionary. The proposed method gives a robust recovery of the coefficients, and a significant denoising effect which can handle up to $100%$ noise-to-signal ratio.
  arXiv  Detail & Related papers  (2022-11-06T14:33:22Z)
• Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty [8.815974147041048]
  We develop a novel framework to identify parsimonious governing equations of nonlinear (spatiotemporal) dynamics from sparse, noisy data with quantified uncertainty. The proposed algorithm is evaluated on multiple nonlinear dynamical systems governed by canonical ordinary and partial differential equations.
  arXiv  Detail & Related papers  (2022-10-14T20:37:36Z)
• Identifiability and Asymptotics in Learning Homogeneous Linear ODE Systems from Discrete Observations [114.17826109037048]
  Ordinary Differential Equations (ODEs) have recently gained a lot of attention in machine learning. theoretical aspects, e.g., identifiability and properties of statistical estimation are still obscure. This paper derives a sufficient condition for the identifiability of homogeneous linear ODE systems from a sequence of equally-spaced error-free observations sampled from a single trajectory.
  arXiv  Detail & Related papers  (2022-10-12T06:46:38Z)
• D-CIPHER: Discovery of Closed-form Partial Differential Equations [80.46395274587098]
  We propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations. We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently.
  arXiv  Detail & Related papers  (2022-06-21T17:59:20Z)
• A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study [68.8204255655161]
  We investigate and compare the performance of several local and global smoothing techniques to a priori denoise the state measurements. We show that, in general, global methods, which use the entire measurement data set, outperform local methods, which employ a neighboring data subset around a local point.
  arXiv  Detail & Related papers  (2022-01-29T23:31:25Z)
• Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data [2.745859263816099]
  A new framework combining deep-learning and integral form is proposed to handle the above-mentioned problems simultaneously. Our proposed algorithm is more robust to noise and more accurate compared with existing methods due to the utilization of integral form.
  arXiv  Detail & Related papers  (2020-11-24T09:18:39Z)
• Weak SINDy For Partial Differential Equations [0.0]
  We extend our Weak SINDy (WSINDy) framework to the setting of partial differential equations (PDEs) The elimination of pointwise derivative approximations via the weak form enables effective machine-precision recovery of model coefficients from noise-free data. We demonstrate WSINDy's robustness, speed and accuracy on several challenging PDEs.
  arXiv  Detail & Related papers  (2020-07-06T16:03:51Z)

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.