• Corpus ID: 202577827

Machine Discovery of Partial Differential Equations from Spatiotemporal Data

@article{Yuan2019MachineDO,
  title={Machine Discovery of Partial Differential Equations from Spatiotemporal Data},
  author={Ye Yuan and Junlin Li and Liang Li and Frank Jiang and Xiuchuan Tang and Fumin Zhang and Sheng Liu and Jorge M. Gonçalves and Henning U. Voss and Xiuting Li and J{\"u}rgen Kurths and Han Ding},
  journal={ArXiv},
  year={2019},
  volume={abs/1909.06730}
}
The study presents a general framework for discovering underlying Partial Differential Equations (PDEs) using measured spatiotemporal data. The method, called Sparse Spatiotemporal System Discovery ($\text{S}^3\text{d}$), decides which physical terms are necessary and which can be removed (because they are physically negligible in the sense that they do not affect the dynamics too much) from a pool of candidate functions. The method is built on the recent development of Sparse Bayesian Learning… 
Sparsistent Model Discovery
TLDR
It is shown that the adaptive Lasso will have more chances of verifying the IRC than the Lasso and it is proposed to integrate it within a deep learning model discovery framework with stability selection and error control.
Fully differentiable model discovery
TLDR
This paper starts by reinterpreting PINNs as multitask models, applying multitask learning using uncertainty, and shows that this leads to a natural framework for including Bayesian regression techniques, and builds a robust model discovery algorithm by using SBL.
Sparsely Constrained Neural Networks for Model Discovery of PDEs
TLDR
A modular framework that combines deep-learning based approaches with an arbitrary sparse regression technique and demonstrates with several examples that this combination facilitates and enhances model discovery tasks.
Can Transfer Neuroevolution Tractably Solve Your Differential Equations?
TLDR
A novel and computationally efficient transfer neuroevolution algorithm that is capable of exploiting relevant experiential priors when solving a new problem, with adaptation to protect against the risk of negative transfer is proposed.
Adaptive support-driven Bayesian reweighted algorithm for sparse signal recovery
TLDR
A restart strategy based on shrinkage-thresholding is developed to conduct adaptive support estimate, which can effectively reduce computation burden and memory demands and outperforms state-of-the-art methods.
Sparse Methods for Automatic Relevance Determination

References

SHOWING 1-10 OF 89 REFERENCES
Data-driven discovery of partial differential equations
TLDR
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
TLDR
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.
Hidden physics models: Machine learning of nonlinear partial differential equations
Robust data-driven discovery of governing physical laws with error bars
  • Sheng Zhang, Guang Lin
  • Computer Science
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2018
TLDR
The data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.
Discovering governing equations from data by sparse identification of nonlinear dynamical systems
TLDR
This work develops a novel framework to discover governing equations underlying a dynamical system simply from data measurements, leveraging advances in sparsity techniques and machine learning and using sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data.
Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations
  • M. Raissi
  • Computer Science
    J. Mach. Learn. Res.
  • 2018
TLDR
This work puts forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time by approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks.
Identification of Nonlinear State-Space Systems From Heterogeneous Datasets
TLDR
A new method to identify nonlinear state-space systems from heterogeneous datasets using a Bayesian learning framework that makes use of “sparse group” priors to allow inference of the sparsest model that can explain the whole set of observed heterogeneous data is proposed.
Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems
TLDR
This work puts forth a machine learning approach for identifying nonlinear dynamical systems from data that combines classical tools from numerical analysis with powerful nonlinear function approximators to distill the mechanisms that govern the evolution of a given data-set.
Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations
We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial
Automated reverse engineering of nonlinear dynamical systems
TLDR
This work introduces for the first time a method that can automatically generate symbolic equations for a nonlinear coupled dynamical system directly from time series data, applicable to any system that can be described using sets of ordinary nonlinear differential equations.
...
...