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Há 6 dias · Data-driven discovery is revolutionizing the modeling, prediction, and control of complex systems. This textbook brings together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science.
- Steven L. Brunton, J. Nathan Kutz
- 2019
Learn how to apply data science methods to model and control complex systems in engineering and physics. This textbook covers machine learning, dynamical systems, optimization, and more with examples and code.
26 de jun. de 2024 · Highlights many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, e.g. turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy.
- Contents
- Preface
- Themes of This Book
- Acknowledgments
- Preface
- How to Use This Book
- Preface
- Most Common Optimization Strategies
- Ax = b. (0.1)
- AT = T . (0.2)
- y = Cx. (0.4)
- X = U V∗ ≈ ̃U ̃ ̃V∗. (0.5)
- VV∗
- Overdetermined and Underdetermined Optimization for Nonlinear Systems
- xk+1 = F(xk). (0.11)
- Ktg = g ◦ Ft ⇒ Ktφ = λφ. (0.12)
- Dimensions
- ̃A AX AY
- Matrices, continued
- Operators, Functions, and Maps
- Most Common Acronyms
- Other Acronyms
- Other Acronyms, continued
Preface Common Optimization Techniques, Equations, Symbols, and Acronyms
This book is about the growing intersection of data-driven methods, applied optimization, and the classical fields of engineering mathematics and mathematical physics. We have been developing this material over a number of years, primarily to educate our advanced undergrad and beginning graduate students from engineering and physical science depart...
There are a number of key themes that have emerged throughout this book. First, many complex systems exhibit dominant low-dimensional patterns in the data, despite the rapidly increasing resolution of measurements and computations. This underlying structure enables efficient sensing, and compact representations for modeling and control. Pattern ext...
We are indebted to many wonderful students, collaborators, and colleagues for valuable feedback, suggestions, and support. We are especially grateful to Joshua Proctor, who was
xi instrumental in the origination of this book and who helped guide much of the framing and organization. We have also benefited from extensive interactions and conversations with Bing Brunton, Igor Mezi ́c, Bernd Noack, and Sam Taira. This work would also not be possible without our many great colleagues and collaborators, with whom we have worke...
Our intended audience includes beginning graduate students, or advanced undergraduates, in engineering and science. As such, the machine learning methods are introduced at a beginning level, whereas we assume students know how to model physical systems with differential equations and simulate them with solvers such as ode45. The diversity of topics...
statistics students who often have limited knowledge of dynamical systems and control. Various courses can be designed from this material, and several example syllabi may be found on the book website; this includes homework, data sets, and code. First and foremost, we want this book to be fun, inspiring, eye-opening, and empowering for young scient...
Least-Squares (discussed in Chapters 1 and 4) minimizes the sum of the squares of the residuals between a given fitting model and data. Linear least-squares, where the residuals are linear in the unknowns, has a closed form solution which can be computed by taking the derivative of the residual with respect to each unknown and setting it to zero. I...
The matrix A ∈ Rp×n and vector b ∈ Rp are generally known, and the vector x ∈ Rn is unknown.
The columns ξk of the matrix T are the eigenvectors of A ∈ Cn×n corresponding to the eigenvalue λk: Aξk = λkξk. The matrix is a diagonal matrix containing these eigenvalues, in the simple case with n distinct eigenvalues.
The vector y ∈ Rp is a measurement of the state x Rn ∈ by the measurement matrix C ∈ Rp×n.
The matrix X ∈ Cn×m may be decomposed into the product of three matrices U ∈ Cn×n, ∈ Cn×m, and V ∈ Cm×m. The matrices U and V are unitary, so that UU∗ = U∗U = In×n and = V∗V = Im×m, where ∗ denotes complex conjugate transpose. The columns of U (resp. V) are orthogonal, called left (resp. right) singular vectors. The matrix contains
decreasing, nonnegative diagonal entries called singular values. Often, X is approximated with a low-rank matrix ̃X = ̃U ̃ ̃V∗, where ̃U and ̃V contain the first r ≪ n columns of U and V, respectively, and ̃ contains the first r × r block of . The matrix ̃U is often denoted in the context of spatial modes, reduced order models, and sensor p...
argmin (f (A, x, b) + λg(x)) x argmin g(x) subject to or
The state of the system at the kth iteration is xk ∈ Rn, and F is a possibly nonlinear mapping. Often, this map defines an iteration forward in time, so that xk = x(kt) ; in this case the flow map is denoted Ft .
The linear Koopman operator Kt advances measurement functions of the state g(x) with the flow Ft. Eigenvalues and eigenvectors of Kt are λ and φ(x), respectively. The operator Kt operates on a Hilbert space of measurements.
K Number of nonzero entries in a K-sparse vector s m Number of data snapshots (i.e., columns of X) n Dimension of the state, x Rn ∈ p Dimension of the measurement or output variable, y Rp q Dimension of the input variable, u Rq ∈ ∈ r Rank of truncated SVD, or other low-rank approximation
Reduced dynamics on r-dimensional POD subspace Matrix representation of linear dynamics on the state x Matrix representation of linear dynamics on the observables y (A, B, C, B) Matrices for continuous-time state-space system (Ad, Bd, Cd, Bd) Matrices for discrete-time state-space system ( ˆA, ˆB, ˆC, ˆB) Matrices for state-space system in new coor...
Matrix of singular values of X, ∈ Rn×m ˆ Matrix of singular values of economy SVD of X, ̃ Matrix of singular values of truncated SVD of X, W Eigenvectors of ̃A Wc Controllability Gramian Wo Observability Gramian X Data matrix, X ∈ Rn×m X′ Time-shifted data matrix, X′ ∈ Rn×m Y Projection of X
Fourier transform Discrete-time dynamical system map Ft Discrete-time flow map of dynamical system through time t Continuous-time dynamical system Gabor transform xx Common Optimization Techniques, Equations, Symbols, and Acronyms
CNN Convolutional neural network DL DMD FFT ODE PCA PDE POD ROM SVD Deep learning Dynamic mode decomposition Fast Fourier transform Ordinary differential equation Principal components analysis Partial differential equation Proper orthogonal decomposition Reduced order model Singular value decomposition
ADM AIC ALM ANN ARMA ARMAX BIC BPOD DMDc CCA CFD CoSaMP CWT DEIM DCT DFT DMDc DNS DWT ECOG eDMD EIM EM EOF ERA ESC GMM HAVOK JL Alternating directions method Akaike information criterion Augmented Lagrange multiplier Artificial neural network Autoregressive moving average Autoregressive moving average with exogenous input Bayesian information crite...
KLT LAD LASSO LDA LQE LQG LQR LTI MIMO MLC MPE mrDMD NARMAX NLS OKID PBH PCP PDE-FIND PDF PID PIV RIP rSVD RKHS RNN RPCA SGD SINDy SISO SRC SSA STFT STLS SVM TICA VAC Karhunen–Loève transform Least absolute deviations Least absolute shrinkage and selection operator Linear discriminant analysis Linear quadratic estimator Linear quadratic Gaussian co...
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Data-Driven Science and Engineering Data-driven discovery is revolutionizing how we model, pre dict, and control complex systems. Now with Python and MATLAB, this textbook trains mathematical scientists and engineers for the next generation of scientiÞc discover y by offering a broad overview of the growing intersection of data-driven methods ...
5 de mai. de 2022 · Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Steven L. Brunton, J. Nathan Kutz. Cambridge University Press, May 5, 2022 - Computers - 590 pages....
Learn how to apply data-driven methods, machine learning, and optimization to complex systems with MATLAB. This book covers topics from introductory to research-level material, with new chapters on reinforcement learning and physics-informed machine learning.
