Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: A case study with the Lorenz 96 model

A novel method, based on the combination of data assimilation and machine learning is introduced. The new hybrid approach is designed for a two-fold scope: (i) emulating hidden, possibly chaotic, dynamics and (ii) predicting their future states. The method consists in applying iteratively a data assimilation step, here an ensemble Kalman filter, and a neural network. Data assimilation is used to optimally combine a surrogate model with sparse noisy data. The output analysis is spatially complete and is used as a training set by the neural network to update the surrogate model. The two steps are then repeated iteratively. Numerical experiments have been carried out using the chaotic 40-variables Lorenz 96 model, proving both convergence and statistical skill of the proposed hybrid approach. The surrogate model shows short-term forecast skill up to two Lyapunov times, the retrieval of positive Lyapunov exponents as well as the more energetic frequencies of the power density spectrum. The sensitivity of the method to critical setup parameters is also presented: the forecast skill decreases smoothly with increased observational noise but drops abruptly if less than half of the model domain is observed. The successful synergy between data assimilation and machine learning, proven here with a low-dimensional system, encourages further investigation of such hybrids with more sophisticated dynamics.

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Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model

Recent progress in machine learning (ML) inspires the idea of improving (or learning) earth system models directly from the observations. Earth sciences already use data assimilation (DA), which underpins decades of progress in weather forecasting. DA and ML have many similarities: they are both inverse methods that can be united under a Bayesian (probabilistic) framework. ML could benefit from approaches used in DA, which has evolved to deal with real observations-these are uncertain, sparsely sampled, and only indirectly sensitive to the processes of interest. DA could also become more like ML and start learning improved models of the earth system, using parameter estimation, or by directly incorporating machine-learnable models. DA follows the Bayesian approach more exactly in terms of representing uncertainty, and in retaining existing physical knowledge, which helps to better constrain the learnt aspects of models. This article makes equivalences between DA and ML in the unifying framework of Bayesian networks. These help illustrate the equivalences between four-dimensional variational (4D-Var) DA and a recurrent neural network (RNN), for example. More broadly, Bayesian networks are graphical representations of the knowledge and processes embodied in earth system models, giving a framework for organising modelling components and knowledge, whether coming from physical equations or learnt from observations. Their full Bayesian solution is not computationally feasible but these networks can be solved with approximate methods already used in DA and ML, so they could provide a practical framework for the unification of the two. Development of all these approaches could address the grand challenge of making better use of observations to improve physical models of earth system processes. This article is part of the theme issue 'Machine learning for weather and climate modelling'.

Learning earth system models from observations: machine learning or data assimilation?

Deep learning for small and big data in psychiatry.

Inverse Methods in Physical Oceanography: Bibliography

Networks of nonlinear systems contain unknown parameters and dynamical degrees of freedom that may not be observable with existing instruments. From observable state variables, we want to estimate the connectivity of a model of such a network and determine the full state of the model at the termination of a temporal observation window during which measurements transfer information to a model of the network. The model state at the termination of a measurement window acts as an initial condition for predicting the future behavior of the network. This allows the validation (or invalidation) of the model as a representation of the dynamical processes producing the observations. Once the model has been tested against new data, it may be utilized as a predictor of responses to innovative stimuli or forcing. We describe a general framework for the tasks involved in the “inverse” problem of determining properties of a model built to represent measured output from physical, biological, or other processes when the ...

A unifying view of synchronization for data assimilation in complex nonlinear networks.

Data assimilation (DA) solves the inverse problem of inferring initial conditions given data and a model. Here we use biophysically motivated Hodgkin-Huxley (HH) models of avian HVCI neurons, experimentally obtained recordings of these neurons, and our data assimilation algorithm to infer the full set of parameters and a minimal set of ionic currents precisely reproducing the observed waveform information. We find many distinct validated sets of parameters selected by our DA method and choice of model. We conclude exploring variations on the inverse problem applied to neurons producing accurate or inaccurate results; by manipulating data presented to the algorithm, varying sample rate and waveform; and by manipulating the model by adding and subtracting ionic currents.

HVC Interneuron Properties from Statistical Data Assimilation

We identify a strong equivalence between neural network based machine learning (ML) methods and the formulation of statistical data assimilation (DA), known to be a problem in statistical physics. DA, as used widely in physical and biological sciences, systematically transfers information in observations to a model of the processes producing the observations. The correspondence is that layer label in the ML setting is the analog of time in the data assimilation setting. Utilizing aspects of this equivalence we discuss how to establish the global minimum of the cost functions in the ML context, using a variational annealing method from DA. This provides a design method for optimal networks for ML applications and may serve as the basis for understanding the success of "deep learning". Results from an ML example are presented. 
When the layer label is taken to be continuous, the Euler-Lagrange equation for the ML optimization problem is an ordinary differential equation, and we see that the problem being solved is a two point boundary value problem. The use of continuous layers is denoted "deepest learning". The Hamiltonian version provides a direct rationale for back propagation as a solution method for the canonical momentum; however, it suggests other solution methods are to be preferred.

Machine Learning as Statistical Data Assimilation.

/pdf/response-to-comment-on-a-unifying-view-of-synchronization-bw9zcba7mw.pdf

Response to "Comment on 'A unifying view of synchronization for data assimilation in complex nonlinear networks'" [Chaos 28, 028101 (2018)].

In statistical data assimilation one seeks the largest maximum of the conditional probability distribution $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$ of model states, $\mathbf{X}$, and parameters,$\mathbf{p}$, conditioned on observations $\mathbf{Y}$ through minimizing the `action', $A(\mathbf{X}) = -\log P(\mathbf{X},\mathbf{p}|\mathbf{Y})$. This determines the dominant contribution to the expected values of functions of $\mathbf{X}$ but does not give information about the structure of $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$ away from the maximum. We introduce a Monte Carlo sampling method, called Strategic Monte Carlo (SMC) sampling, for estimating $P(\mathbf{X}, \mathbf{p}|\mathbf{Y})$ in the neighborhood of its largest maximum to remedy this limitation. SMC begins with a systematic variational annealing (VA) procedure for finding the smallest minimum of $A(\mathbf{X})$. SMC generates accurate estimates for the mean, standard deviation and other higher moments of $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$. Additionally, the random search allows for an understanding of any multimodal structure that may underly the dynamics of the problem. SMC generates a gaussian probability control term based on the paths determined by VA to minimize a cost function $A(\mathbf{X},\mathbf{p})$. This probability is sampled during the Monte Carlo search of the cost function to constrain the search to high probability regions of the surface thus substantially reducing the time necessary to sufficiently explore the space.

Strategic Monte Carlo Methods for State and Parameter Estimation in High Dimensional Nonlinear Problems

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