Newtonian Nonlinear Dynamics for Complex Linear and Optimization Problems
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In , Noack et al. This mean-field model exhibits a stable limit cycle corresponding to von Karman vortex shedding, and an unstable equilibrium corresponding to a low-drag condition. Starting near this equilibrium, the flow unwinds up the slow manifold toward the limit cycle. In 51 , Loiseau and Brunton showed that this flow may be modeled by a nonlinear oscillator with state-dependent damping, making it amenable to the continuous spectrum analysis. The resulting eigenfunctions are shown in Fig. Learned Koopman eigenfunctions for the mean-field model of fluid flow past a circular cylinder at Reynolds number The Koopman model is able to reconstruct both given only the initial condition.
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These continuously varying eigenvalues are shown in Supplementary Fig. This is consistent with the data-driven model of Loiseau and Brunton Although we only show the ability of the model to predict the future state in Fig. Thus, it is possible to capture nonlinear transients, as long as these are sufficiently represented in the training data. In summary, we have employed powerful deep learning approaches to identify and represent coordinate transformations that recast strongly nonlinear dynamics into a globally linear framework. Our approach is designed to discover eigenfunctions of the Koopman operator, which provide an intrinsic coordinate system to linearize nonlinear systems, and have been notoriously difficult to identify and represent using alternative methods.
Building on a deep auto-encoder framework, we enforce additional constraints and loss functions to identify Koopman eigenfunctions where the dynamics evolve linearly. Moreover, we generalize this framework to include a broad class of nonlinear systems that exhibit a continuous eigenvalue spectrum, where a continuous range of frequencies is observed. Continuous-spectrum systems are notoriously difficult to analyze, especially with Koopman theory, and naive learning approaches require asymptotic expansions in terms of higher order harmonics of the fundamental frequency, leading to unwieldy models.
In contrast, we utilize an auxiliary network to parametrize and identify the continuous frequency, which then parameterizes a compact Koopman model on the auto-encoder coordinates. Thus, our deep neural network models remain both parsimonious and interpretable, merging the best of neural network representations and Koopman embeddings. In most deep learning applications, although the basic architecture is extremely general, considerable expert knowledge and intuition is typically used in the training process and in designing loss functions and constraints.
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Throughout this paper, we have also used physical insight and intuition from asymptotic theory and continuous spectrum dynamical systems to guide the design of parsimonious Koopman embeddings. There are many ongoing challenges and promising directions that motivate future work. First, there are still several limitations associated with deep learning, including the need for vast and diverse data and extensive computation to train models This training may be considered a one-time upfront cost, and deep learning frameworks, such as TensorFlow parallelize the training on GPUs and across GPUs 54 ; further, there is ongoing work to improve the scalability Even more concerning is the dubious generalizability and interpretability of the resulting models, as deep learning architectures may be viewed as sophisticated interpolation engines with limited ability to extrapolate beyond the training data This work attempts to promote interpretability by forcing the network to have physical meaning in the context of Koopman theory, although the issue with generalizability still requires sufficient volumes and diversity of training data.
There are also more specific limitations to the current proposed architecture, foremost, choosing the dimension of the autoencoder coordinates, y. Continued effort will be required to automatically detect the dimension of the intrinsic coordinates and to classify spectra e. It will be important to extend these methods to higher-dimensional examples with more complex energy spectra, as the examples considered here are relatively low-dimensional. Fortunately, with sufficient data, deep learning architectures are able to learn incredibly complex representations, so the prospects for scaling these methods to larger systems is promising.
The use of deep learning in physics and engineering is increasing at an incredible rate, and this trend is only expected to accelerate. Nearly every field of science is revisiting challenging problems of central importance from the perspective of big data and deep learning. With this explosion of interest, it is imperative that we as a community seek machine learning models that favor interpretability and promote physical insight and intuition.
In this challenge, there is a tremendous opportunity to gain new understanding and insight by applying increasingly powerful techniques to data. For example, discovering Koopman eigenfunctions will result in new symmetries and conservation laws, as conserved eigenfunctions are related to conservation laws via a generalized Noether's theorem. It will also be important to apply these techniques to increasingly challenging problems, such as turbulence, epidemiology, and neuroscience, where data is abundant and models are needed.
The goal is to model these systems with a small number of coupled nonlinear oscillators using similar parameterized Koopman embeddings. Finally, the use of deep learning to discover Koopman eigenfunctions may enable transformative advances in the nonlinear control of complex systems. All of these future directions will be facilitated by more powerful network representations.
For each initial condition, we solve the differential equations for some time span. Note that for the network to capture transient behavior as in the first and last example, it is important to include enough samples of transients in the training data.
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These ranges are chosen to sample the pendulum in the full phase space where the pendulum approaches having an infinite period. These limits are chosen to include the dynamics on the slow manifold covered by the previous dataset, as well as trajectories that begin off the slow manifold.
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All of our code is available online at github. In our experiments, training was significantly faster with ReLU as the activation function than with sigmoid. The input to the auxiliary network is y , and it outputs the parameters for the eigenvalues of K. For example, for the fluid flow problem off the attractor, we have three eigenfunctions. However, for simplicity, we implement this as two separate auxiliary networks, one for the complex conjugate pair of eigenvalues and one for the the real eigenvalue. More specifically:. The integer S p is a hyperparameter for how many steps to check in the prediction loss.
However, on Hamiltonian systems, such as the pendulum, the eigenvalues are constant along each trajectory. In order to demonstrate that this specialized knowledge is not necessary, we use the more general case for all of our datasets, including the pendulum. This distribution was suggested in ref.
Each bias vector b is initialized to 0. The learning rate for the Adam optimizer is 0. We also use early stopping; for each model, at the end of training, we resume the step with the lowest validation error. There are many design choices in deep learning, so we use hyperparameter tuning, as described in ref. For each dynamical system, we train multiple models in a random search of hyperparameter space and choose the one with the lowest validation error.
Each model is also initialized with different random weights. All data generated during this study can be reconstructed using the code available at github. Guckenheimer, J. Cross, M. Pattern formation outside of equilibrium. Dullerud, G. Texts in Applied Mathematics. Springer-Verlag: New York, Koopman, B.
Hamiltonian systems and transformation in Hilbert space. Natl Acad.
USA 17 , — USA 18 , — Comparison of systems with complex behavior. D , — Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. Applied Koopmanism a.
Newtonian Nonlinear Dynamics for Complex Linear and Optimization Problems
Chaos 22 , Mezic, I. Analysis of fluid flows via spectral properties of the Koopman operator. Schmid, P. Dynamic mode decomposition of numerical and experimental data. Fluid Mech.
معلومات عن المنتج
Rowley, C. Spectral analysis of nonlinear flows. Kutz, J. Hubel, D.