# 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} }

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…

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## References

SHOWING 1-10 OF 49 REFERENCES

### Equations of Motion from a Data Series

- Computer ScienceComplex 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 ScienceProceedings 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.

- MathematicsPhysical 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

- Environmental Science
- 2002

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

- Mathematics
- 1963

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

- PhysicsScience
- 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

- Mathematics, Computer Science
- 2006

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 ScienceIEEE 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.