Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

https://www.pnas.org/content/pnas/113/15/3932.full.pdf

Discovering governing equations from data by sparse identification of nonlinear dynamical systems

8. Subset Selection in Regression (Monographs on Statistics and Applied Probability, no. 40). By A. J. Miller. ISBN 0 412 35380 6. Chapman and Hall, London, 1990. 240 pp. £25.00.

Subset Selection in Regression

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The 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. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg–de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable.

Data-driven discovery of partial differential equations.

Hidden physics models: Machine learning of nonlinear partial differential equations

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Mathematical Methods Of Statistics

Optimal power scheduling for distributed detection in a Gaussian sensor network is addressed for both independent and correlated observations. We assume amplify-and-forward local processing at each node. The wireless link between sensors and the fusion center is assumed to undergo fading and coefficients are assumed to be available at the transmitting sensors. The objective is to minimize the total network power to achieve a desired fusion error probability at the fusion center. For i.i.d. observations, the optimal power allocation is derived analytically in closed form. When observations are correlated, first, an easy to optimize upper bound is derived for sufficiently small correlations and the power allocation scheme is derived accordingly. Next, an evolutionary computation technique based on particle swarm optimization is developed to find the optimal power allocation for arbitrary correlations. The optimal power scheduling scheme suggests that the sensors with poor observation quality and bad channels should be inactive to save the total power expenditure of the system. It is shown that the probability of fusion error performance based on the optimal power allocation scheme outperforms the uniform power allocation scheme especially when either the number of sensors is large or the local observation quality is good.

/pdf/optimal-power-scheduling-for-correlated-data-fusion-in-pclw7qqqa1.pdf

Optimal Power Scheduling for Correlated Data Fusion in Wireless Sensor Networks via Constrained PSO

Compressive sensing is used to compress and reconstruct a turbulent-flow particle image velocimetry database over a NACA 4412 airfoil. The spatial velocity data at a given time are sufficiently sparse in the discrete cosine transform basis, and the feasibility of compressive sensing for velocity data reconstruction is demonstrated. Application of the proper orthogonal decomposition/principal component analysis on the dataset works better than the compressive-sensing-based reconstruction approach with discrete cosine transform as the basis in terms of the reconstruction error, although the performance gap between the two schemes is not significant. Using the proper orthogonal decomposition/principal component analysis as the sparsifying basis, compressive-sensing-based velocity reconstruction is implemented, which outperformed discrete cosine transform. Compressive sensing preprocessing (filtering) with discrete cosine transform as the basis is applied to a reduced number of particle image velocimetry snap...

/pdf/low-dimensional-approach-for-reconstruction-of-airfoil-data-17qwg0ex7d.pdf

Low-Dimensional Approach for Reconstruction of Airfoil Data via Compressive Sensing

Most of the recent compressive sensing (CS) literature has focused on sparse signal recovery based on compressive measurements. However, exact signal recovery may not be required in certain signal processing applications such as in inference problems. In this paper, we provide performance limits of classification of sparse as well as not necessarily sparse signals based on compressive measurements. When signals are not necessarily sparse, we show that Kullback-Leibler and Chernoff distances between two probability density functions under any two hypotheses are preserved up to a factor of M/N with M(<;N)-length compressive measurements compared to that with N-length original measurements when the pdfs of the original-length observation vectors exhibit certain properties. These results are used to quantify the performance limits in terms of upper and lower bounds on the probability of error in signal classification with M-length compressive measurements. When the signals of interest are sparse in the standard canonical basis, performance limits are derived in terms of lower bounds on the probability of error in classifying sparse signals with any classification rule.

Performance Limits of Compressive Sensing-Based Signal Classification

We address the problem of joint sparsity pattern recovery based on low dimensional multiple measurement vectors (MMVs) in resource constrained distributed networks. We assume that distributed nodes observe sparse signals which share the same sparsity pattern and each node obtains measurements via a low dimensional linear operator. When the measurements are collected at distributed nodes in a communication network, it is often required that joint sparse recovery be performed under inherent resource constraints such as communication bandwidth and transmit/processing power. 
We present two approaches to take the communication constraints into account while performing common sparsity pattern recovery. First, we explore the use of a shared multiple access channel (MAC) in forwarding observations residing at each node to a fusion center. With MAC, while the bandwidth requirement does not depend on the number of nodes, the fusion center has access to only a linear combination of the observations. We discuss the conditions under which the common sparsity pattern can be estimated reliably. Second, we develop two collaborative algorithms based on Orthogonal Matching Pursuit (OMP), to jointly estimate the common sparsity pattern in a decentralized manner with a low communication overhead. In the proposed algorithms, each node exploits collaboration among neighboring nodes by sharing a small amount of information for fusion at different stages in estimating the indices of the true support in a greedy manner. Efficiency and effectiveness of the proposed algorithms are demonstrated via simulations along with a comparison with the most related existing algorithms considering the trade-off between the performance gain and the communication overhead.

OMP based joint sparsity pattern recovery under communication constraints

In this paper, we consider the problem of recovering an unknown sparse matrix X from the matrix sketch Y = AX B^T. The dimension of Y is less than that of X, and A and B are known matrices. This problem can be solved using standard compressive sensing (CS) theory after converting it to vector form using the Kronecker operation. In this case, the measurement matrix assumes a Kronecker product structure. However, as the matrix dimension increases the associated computational complexity makes its use prohibitive. We extend two algorithms, fast iterative shrinkage threshold algorithm (FISTA) and orthogonal matching pursuit (OMP) to solve this problem in matrix form without employing the Kronecker product. While both FISTA and OMP with matrix inputs are shown to be equivalent in performance to their vector counterparts with the Kronecker product, solving them in matrix form is shown to be computationally more efficient. We show that the computational gain achieved by FISTA with matrix inputs over its vector form is more significant compared to that achieved by OMP.

Thakshila Wimalajeewa

Papers

Optimal Power Scheduling for Correlated Data Fusion in Wireless Sensor Networks via Constrained PSO

Low-Dimensional Approach for Reconstruction of Airfoil Data via Compressive Sensing

Performance Limits of Compressive Sensing-Based Signal Classification

OMP based joint sparsity pattern recovery under communication constraints

Recovery of Sparse Matrices via Matrix Sketching.