Paul J. Rozdeba

Alexander J. A. Ty, Zheng Fang, Rivver A. Gonzalez et al.

Tasking machine learning to predict segments of a time series requires estimating the parameters of a ML model with input/output pairs from the time series. Using the equivalence between statistical data assimilation and supervised machine learning, we revisit this task. The training method for the machine utilizes a precision annealing approach to identifying the global minimum of the action (-log[P]). In this way we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series $s(t_n); t_n = t_0 + n Δt$ and using methods of nonlinear time series analysis show how to produce a $D_E > 1$ dimensional time delay embedding space in which the time series has no false neighbors as does the observed $s(t_n)$ time series. In that $D_E$-dimensional space we explore the use of feed forward multi-layer perceptrons as network models operating on $D_E$-dimensional input and producing $D_E$-dimensional outputs.

H. D. I. Abarbanel, P. J. Rozdeba, S. Shirman

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.

Henry Abarbanel, Paul Rozdeba, Sasha Shirman

We formulate a strong equivalence between machine learning, artificial intelligence methods and the formulation of statistical data assimilation as used widely in physical and biological sciences. The correspondence is that layer number in the artificial network setting is the analog of time in the data assimilation setting. Within the discussion of this equivalence we show that adding more layers (making the network deeper) is analogous to adding temporal resolution in a data assimilation framework. How one can find a candidate for the global minimum of the cost functions in the machine learning context using a method from data assimilation is discussed. Calculations on simple models from each side of the equivalence are reported. Also discussed is a framework in which the time or layer label is taken to be continuous, providing a differential equation, the Euler-Lagrange equation, which shows that the problem being solved is a two point boundary value problem familiar in the discussion of variational methods. The use of continuous layers is denoted "deepest learning". These problems respect a symplectic symmetry in continuous time/layer phase space. Both Lagrangian versions and Hamiltonian versions of these problems are presented. Their well-studied implementation in a discrete time/layer, while respected the symplectic structure, is addressed. The Hamiltonian version provides a direct rationale for back propagation as a solution method for the canonical momentum.