Learning Temporal Dependence from Time-Series Data with Latent Variables

We consider the setting where a collection of time series, modeled as random processes, evolve in a causal manner, and one is interested in learning the graph governing the relationships of these processes. A special case of wide interest and applicability is the setting where the noise is Gaussian and relationships are Markov and linear. We study this setting with two additional features: firstly, each random process has a hidden (latent) state, which we use to model the internal memory possessed by the variables (similar to hidden Markov models). Secondly, each variable can depend on its latent memory state through a random lag (rather than a fixed lag), thus modeling memory recall with differing lags at distinct times. Under this setting, we develop an estimator and prove that under a genericity assumption, the parameters of the model can be learned consistently. We also propose a practical adaption of this estimator, which demonstrates significant performance gains in both synthetic and real-world datasets

Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations

This paper introduces tensorial calculus techniques in the framework of Proper Orthogonal Decomposition (POD) to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree $p$. Such nonlinear terms have an on-line complexity of $\mathcal{O}(k^{p+1})$, where $k$ is the dimension of POD basis, and therefore is independent of full space dimension. However it is efficient only for quadratic nonlinear terms since for higher nonlinearities standard POD proves to be less time consuming once the POD basis dimension $k$ is increased. Numerical experiments are carried out with a two dimensional shallow water equation (SWE) test problem to compare the performance of tensorial POD, standard POD, and POD/Discrete Empirical Interpolation Method (DEIM). Numerical results show that tensorial POD decreases by $76\times$ times the computational cost of the on-line stage of standard POD for configurations using more than $300,000$ model variables. The tensorial POD SWE model was only $2-8\times$ slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM shallow water equation model to compute its off-line stage faster than the standard and tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table

Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes

The application of Stochastic Differential Equations (SDEs) to the analysis of temporal data has attracted increasing attention, due to their ability to describe complex dynamics with physically interpretable equations. In this paper, we introduce a non-parametric method for estimating the drift and diffusion terms of SDEs from a densely observed discrete time series. The use of Gaussian processes as priors permits working directly in a function-space view and thus the inference takes place directly in this space. To cope with the computational complexity that requires the use of Gaussian processes, a sparse Gaussian process approximation is provided. This approximation permits the efficient computation of predictions for the drift and diffusion terms by using a distribution over a small subset of pseudo-samples. The proposed method has been validated using both simulated data and real data from economy and paleoclimatology. The application of the method to real data demonstrates its ability to capture the behaviour of complex systems

Calculation of Generalized Polynomial-Chaos Basis Functions and Gauss Quadrature Rules in Hierarchical Uncertainty Quantification

Stochastic spectral methods are efficient techniques for uncertainty quantification. Recently they have shown excellent performance in the statistical analysis of integrated circuits. In stochastic spectral methods, one needs to determine a set of orthonormal polynomials and a proper numerical quadrature rule. The former are used as the basis functions in a generalized polynomial chaos expansion. The latter is used to compute the integrals involved in stochastic spectral methods. Obtaining such information requires knowing the density function of the random input {\it a-priori}. However, individual system components are often described by surrogate models rather than density functions. In order to apply stochastic spectral methods in hierarchical uncertainty quantification, we first propose to construct physically consistent closed-form density functions by two monotone interpolation schemes. Then, by exploiting the special forms of the obtained density functions, we determine the generalized polynomial-chaos basis functions and the Gauss quadrature rules that are required by a stochastic spectral simulator. The effectiveness of our proposed algorithm is verified by both synthetic and practical circuit examples.Comment: Published by IEEE Trans CAD in May 201

Application of Sparse Identification of Nonlinear Dynamics for Physics-Informed Learning

Advances in machine learning and deep neural networks has enabled complex engineering tasks like image recognition, anomaly detection, regression, and multi-objective optimization, to name but a few. The complexity of the algorithm architecture, e.g., the number of hidden layers in a deep neural network, typically grows with the complexity of the problems they are required to solve, leaving little room for interpreting (or explaining) the path that results in a specific solution. This drawback is particularly relevant for autonomous aerospace and aviation systems, where certifications require a complete understanding of the algorithm behavior in all possible scenarios. Including physics knowledge in such data-driven tools may improve the interpretability of the algorithms, thus enhancing model validation against events with low probability but relevant for system certification. Such events include, for example, spacecraft or aircraft sub-system failures, for which data may not be available in the training phase. This paper investigates a recent physics-informed learning algorithm for identification of system dynamics, and shows how the governing equations of a system can be extracted from data using sparse regression. The learned relationships can be utilized as a surrogate model which, unlike typical data-driven surrogate models, relies on the learned underlying dynamics of the system rather than large number of fitting parameters. The work shows that the algorithm can reconstruct the differential equations underlying the observed dynamics using a single trajectory when no uncertainty is involved. However, the training set size must increase when dealing with stochastic systems, e.g., nonlinear dynamics with random initial conditions

An adaptive orthogonal search algorithm for model subset selection and non-linear system identification

A new adaptive orthogonal search (AOS) algorithm is proposed for model subset selection and non-linear system identification. Model structure detection is a key step in any system identification problem. This consists of selecting significant model terms from a redundant dictionary of candidate model terms, and determining the model complexity (model length or model size). The final objective is to produce a parsimonious model that can well capture the inherent dynamics of the underlying system. In the new AOS algorithm, a modified generalized cross-validation criterion, called the adjustable prediction error sum of squares (APRESS), is introduced and incorporated into a forward orthogonal search procedure. The main advantage of the new AOS algorithm is that the mechanism is simple and the implementation is direct and easy, and more importantly it can produce efficient model subsets for most non-linear identification problems