• Corpus ID: 118625096

# Reconstruction of Ordinary Differential Equations From Time Series Data

@article{Mai2016ReconstructionOO,
title={Reconstruction of Ordinary Differential Equations From Time Series Data},
author={Manuel Mai and Mark D. Shattuck and Corey S. O’Hern},
journal={arXiv: Data Analysis, Statistics and Probability},
year={2016}
}
• Published 18 May 2016
• Computer Science
• arXiv: Data Analysis, Statistics and Probability
We develop a numerical method to reconstruct systems of ordinary differential equations (ODEs) from time series data without {\it a priori} knowledge of the underlying ODEs using sparse basis learning and sparse function reconstruction. We show that employing sparse representations provides more accurate ODE reconstruction compared to least-squares reconstruction techniques for a given amount of time series data. We test and validate the ODE reconstruction method on known 1D, 2D, and 3D systems…
5 Citations

## Figures from this paper

### Predicting dynamical system evolution with residual neural networks

• Computer Science
Keldysh Institute Preprints
• 2019
It is shown how by training neural networks with ResNet-like architecture on the solution samples, models can be developed to predict the ODE system solution further in time and the predicted solution remains stable for much longer times than for other currently known models.

### Neural ordinary differential equations for ecological and evolutionary time‐series analysis

• Environmental Science
Methods in Ecology and Evolution
• 2020
Inferring the functional shape of ecological and evolutionary processes from time‐series data can be challenging because processes are often not describable with simple equations. The dynamical

### Fast fitting of neural ordinary differential equations by Bayesian neural gradient matching to infer ecological interactions from time series data

• Computer Science
• 2022
A fast NODE fitting method, Bayesian neural gradient matching (BNGM), which relies on interpolating time series with neural networks, and fitting NODEs to the interpolated dynamics with Bayesian regularisation, and provides accurate estimates of ecological interactions in the artificial system.

### Data-Driven discovery of governing physical laws and their parametric dependencies in engineering, physics and biology

• Computer Science
2017 IEEE 7th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)
• 2017
A regression method based upon group sparsity that is capable of discovering parametrized governing dynamical equations of motion of a given system by time series measurements is proposed, giving a promising new technique for disambiguating governing equations from simple parametric dependencies in physical, biological and engineering systems.

### Flexible Model Induction through Heuristic Process Discovery

• Computer Science
AAAI
• 2017
This paper describes FPM, a system that implementsuctive process modeling by composing knowledge about algebraic rate expressions and about conceptual processes like predation and remineralization in ecology, and compares its failure-driven approach with a naive scheme that generates all possible processes at the outset.

## References

SHOWING 1-10 OF 49 REFERENCES

### Equations of Motion from a Data Series

• Computer Science
Complex Syst.
• 1987
A method to reconstruct the deterministic portion of the equations of motion directly from a data series to represent a vast reduction of a chaotic data set’s observed complexity to a compact, algorithmic specification is described.

### Automated reverse engineering of nonlinear dynamical systems

• Computer Science
Proceedings of the National Academy of Sciences
• 2007
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.

### Fitting ordinary differential equations to chaotic data.

• BaakeBock
• Mathematics
Physical review. A, Atomic, molecular, and optical physics
• 1992
It is claimed that the problem of estimating parameters in systems of ordinary differential equations which give rise to chaotic time series is naturally tackled by boundary value problem methods and Lyapunov exponents can be computed accurately from time series much shorter than those required by previous methods.

### The development of chaotic advection

The concept of chaotic advection was developed some twenty years ago as an outgrowth of work on advection by interacting point vortices. The term “chaotic advection” was first introduced in the title

### Synchronization of Lorenz-based chaotic circuits with applications to communications

• Computer Science
• 1993
An analogy between synchronization in chaotic systems, nonlinear observers for deterministic systems, and state estimation in probabilistic systems is established and the performance of the Lorenz SCS is compared to an extended Kalman filter for providing state estimates when the measurement consists of a single noisy transmitter component.

### Deterministic nonperiodic flow

Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with

### Distilling Free-Form Natural Laws from Experimental Data

• Physics
Science
• 2009
This work proposes a principle for the identification of nontriviality, and demonstrated this approach by automatically searching motion-tracking data captured from various physical systems, ranging from simple harmonic oscillators to chaotic double-pendula, and discovered Hamiltonians, Lagrangians, and other laws of geometric and momentum conservation.

### For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution

The techniques include the use of random proportional embeddings and almost‐spherical sections in Banach space theory, and deviation bounds for the eigenvalues of random Wishart matrices.

### Stable recovery of sparse overcomplete representations in the presence of noise

• Computer Science
IEEE Transactions on Information Theory
• 2006
This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system and shows that similar stability is also available using the basis and the matching pursuit algorithms.