Weak SINDy For Partial Differential Equations

  title={Weak SINDy For Partial Differential Equations},
  author={Daniel A. Messenger and David M. Bortz},
  journal={Journal of computational physics},

Automatic differentiation to simultaneously identify nonlinear dynamics and extract noise probability distributions from data

A variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al is developed, which can learn a diversity of probability distributions for the measurement noise, including Gaussian, uniform, Gamma, and Rayleigh distributions.

Model discovery in the sparse sampling regime

This work investigates how deep learning can improve model discovery of partial differential equations when the spacing between sensors is large and the samples are not placed on a grid, and shows how leveraging physics informed neural network interpolation and automatic differentiation, allow to better fit the data and its spatiotemporal derivatives, compared to more classic spline interpolations and numerical differentiation techniques.

Model selection of chaotic systems from data with hidden variables using sparse data assimilation.

This work presents a method combining variational annealing-a technique previously used for parameter estimation in chaotic systems with hidden variables-with sparse-optimization methods to perform model identification for chaotic system with unmeasured variables.

Detection and characterization of chemotaxis without cell tracking

A novel method for the analysis of time series of positional data generated from realizations of agent-based processes, which can infer the strength and range of attraction or repulsion exerted on agents, as in chemotaxis is presented.

WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming

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 for some equations.

Asymptotic consistency of the WSINDy algorithm in the limit of continuum data

It is shown that in general the WSINDy estimator is only conditionally asymptotically consistent, yielding discovery of spurious terms with probability one if the noise level is above some critical threshold and the nonlinearities exhibit sufficiently fast growth.

Online Weak-form Sparse Identification of Partial Differential Equations

The core of the method combines a weakform discretization of candidate PDEs with an online proximal gradient descent approach to the sparse regression problem, and does not regularize the `0-pseudo-norm, finding that directly applying its proximal operator leads to efficient online system identification from noisy data.

Learning Mean-Field Equations from Particle Data Using WSINDy

Convergence of weak-SINDy Surrogate Models

An in-depth error analysis for surrogate models generated by a variant of the Sparse Identification of Nonlinear Dynamics (SINDy) method is given and it is shown that the surrogate dynamics converges towards the true dynamics.



Weak SINDy: Galerkin-Based Data-Driven Model Selection

A weak formulation and discretization of the system discovery problem from noisy measurement data that combines the ease of implementation of the SINDy algorithm with the natural noise-reduction of integration to arrive at a more robust and user-friendly method of sparse recovery that correctly identifies systems in both small- noise and large-noise regimes.

IDENT: Identifying Differential Equations with Numerical Time Evolution

A new direction based on the fundamental convergence principle of numerical PDE schemes is proposed, and a performance guarantee is established based on an incoherence property of Lasso to validate and correct the results by time evolution error.

Data-Driven Deep Learning of Partial Differential Equations in Modal Space

Sparse Identification of Nonlinear Dynamical Systems via Reweighted 𝓁s1-regularized Least Squares

Sparse Identification of Truncation Errors

Learning partial differential equations for biological transport models from noisy spatio-temporal data

It is shown that the ANN methodology outperforms previous denoising methods, including finite differences and both local and global polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model.

Using noisy or incomplete data to discover models of spatiotemporal dynamics.

This work presents an alternative and rather general approach that addresses sparse regression difficulty by using a weak formulation of the problem, which allows accurate reconstruction of PDEs involving high-order derivatives, such as the Kuramoto-Sivashinsky equation, from data with a considerable amount of noise.

Robust data-driven discovery of governing physical laws with error bars

  • Sheng ZhangGuang Lin
  • Computer Science
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2018
The data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.

Data-driven discovery of partial differential equations

The sparse regression method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation.

Learning partial differential equations via data discovery and sparse optimization

  • H. Schaeffer
  • Computer Science
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2017
This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data, which uses sparse optimization in order to perform feature selection and parameter estimation.