Pityriasis lichenoides et varioliformis acuta (PLEVA), or Mucha-Habermann disease (MHD), is a cutaneous disorder evident with crops of erythematous macules and papules, usually on the trunk and flexural areas of the extremities. Its etiology remains unknown. PLEVA is speculated to be an inflammatory reaction triggered by certain infectious agents, an inflammatory response secondary to T-cell dyscrasia, or an immune complex-mediated hypersensitivity. Histologic examination of a skin biopsy specimen is the standard for the identification of PLEVA, but definitive diagnosis may be difficult. Apart from the febrile ulcerative variant, which may be fatal, PLEVA tends to be self-limited in its course. Treatment is targeted mainly at the symptomatic relief of pruritus.

We formulate an equivalence between machine learning and the formulation of statistical data assimilation as used widely in physical and biological sciences. The correspondence is that layer number in a feedforward artificial network setting is the analog of time in the data assimilation setting. This connection has been noted in the machine learning literature. We add a perspective that expands on how methods from statistical physics and aspects of Lagrangian and Hamiltonian dynamics play a role in how networks can be trained and designed. 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. Extending this equivalence to recurrent networks is also discussed. We explore 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. Calculations on simple models from both sides 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 and its boundary conditions, as a necessary condition for a minimum of the cost function. This 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 layer phase space. Both Lagrangian versions and Hamiltonian versions of these problems are presented. Their well-studied implementation in a discrete time/layer, while respecting the symplectic structure, is addressed. The Hamiltonian version provides a direct rationale for backpropagation as a solution method for a certain two-point boundary value problem.

In statistical data assimilation one evaluates the conditional expected values, conditioned on measurements, of interesting quantities on the path of a model through observation and prediction windows. This often requires working with very high dimensional integrals in the discrete time descriptions of the observations and model dynamics, which become functional integrals in the continuous-time limit. Two familiar methods for performing these integrals include (1) Monte Carlo calculations and (2) variational approximations using the method of Laplace plus perturbative corrections to the dominant contributions. We attend here to aspects of the Laplace approximation and develop an annealing method for locating the variational path satisfying the Euler-Lagrange equations that comprises the major contribution to the integrals. This begins with the identification of the minimum action path starting with a situation where the model dynamics is totally unresolved in state space, and the consistent minimum of the variational problem is known. We then proceed to slowly increase the model resolution, seeking to remain in the basin of the minimum action path, until a path that gives the dominant contribution to the integral is identified. After a discussion of some general issues, we give examples of the assimilation process for some simple, instructive models from the geophysical literature. Then we explore a slightly richer model of the same type with two distinct time scales. This is followed by a model characterizing the biophysics of individual neurons.

Abstract. Data assimilation transfers information from an observed system to a physically based model system with state variables x(t). The observations are typically noisy, the model has errors, and the initial state x(t 0 ) is uncertain: the data assimilation is statistical. One can ask about expected values of functions G(X) on the path X = {x(t 0 ), . . ., x(t m )} of the model state through the observation window t n = {t 0 , . . ., t m }. The conditional (on the measurements) probability distribution P (X) = exp[−A 0 (X)] determines these expected values. Variational methods using saddle points of the "action" A 0 (X), known as 4DVar (Talagrand and Courtier, 1987;Evensen, 2009), are utilized for estimating G(X) . In a path integral formulation of statistical data assimilation, we consider variational approximations in a realization of the action where measurement errors and model errors are Gaussian. We (a) discuss an annealing method for locating the path X 0 giving a consistent minimum of the action A 0 (X 0 ), (b) consider the explicit role of the number of measurements at each t n in determining A 0 (X 0 ), and (c) identify a parameter regime for the scale of model errors, which allows X 0 to give a precise estimate of G(X 0 ) with computable, small higher-order corrections.

The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems.