The textbook Kalman Filter (LKF) seeks to estimate the state of a linear system based on having two things in hand: a.) a reasonable state-space model of the underlying process and its noise components; b.) imperfect (noisy) measurements obtained from the process via one or more sensors. The LKF approach results in a predictor-corrector algorithm which can be applied recursively to correct predictions from the state model so as to yield posterior estimates of the current process state, as new sensor data are made available. The LKF can be shown to be optimal in a Gaussian setting and is eminently useful in practical settings when the models and measurements are stochastic and non-stationary. Numerous extensions of the KF filter have been proposed for the non-linear problem, such as extended Kalman Filters (EKF) and ‘ensemble’ filters (EnKF). Proofs of optimality are difficult to obtain but for many problems where the ‘physics’ is of modest complexity EKF’s yield algorithms which function well in a practical sense; the EnKF also shows promise but is limited by the requirement for sampling the random processes. In multi-physics systems, for example, several complications arise, even beyond non-Gaussianity. On the one hand, multi-physics effects may include multi-scale responses and path dependency, which may be poorly sampled by a sensor suite (tending to favor low gains). One the other hand, as more multi-physics effects are incorporated into a model, the model itself becomes a less and less perfect model of reality (tending to favor high gains). For reasons such as these suitable estimates of the joint system response are difficult to obtain, as are corresponding joint estimates of the sensor ensemble. This paper will address these issues in a two-fold way — first by a generalized process model representation based on regularized stochastic non-linear networks (Snn), and second by transformation of the process itself by an adaptive low-dimensional subspace in which the update step on the residual can be performed in a space commensurate with the available information content of the process and measured response.

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