Inferring the source information of greenhouse gases, such as methane, from spatially sparse sensor observations is an essential element in mitigating climate change. While it is well understood that the complex behavior of the atmospheric dispersion of such pollutants is governed by the Advection-Diffusion equation, it is difficult to directly apply the governing equations to identify the source location and magnitude (inverse problem) because of the spatially sparse and noisy observations, i.e., the pollution concentration is known only at the sensor locations and sensors sensitivity is limited. Here, we develop a multi-task learning framework that can provide high-fidelity reconstruction of the concentration field and identify emission characteristics of the pollution sources such as their location, emission strength, etc. from sparse sensor observations. We demonstrate that our proposed framework is able to achieve accurate reconstruction of the methane concentrations from sparse sensor measurements as well as precisely pin-point the location and emission strength of these pollution sources.

For large scale monitoring of the environment, the number of possible pollution sources can be larger than the number of sensors. For optimal sensor placement under various wind fields in source inversion problems, this paper proposes a framework under non-Gaussian priors for the detection and inversion estimate of emission rates. The optimization framework with non-Gaussian prior utilizes a bi-level optimization expression with inner quadratic programming. The proposed truncated Gaussian prior is to incorporate non-negativity of emission rates, but it poses a challenge in optimization. We preliminarily investigate the bi-level optimization with a Gaussian plume model example. The Karush–Kuhn–Tucker conditions of the inner quadratic programming are considered for solving the bi-level optimization. The efficiency of the proposed optimization framework is demonstrated by numerical results to optimally place sensors and quantify emission rates.

Model-free identification of a nonlinear dynamical system from the noisy observations is of current interest due to its direct relevance to many applications in Industry 4.0. Making a prediction of such noisy time series constitutes a problem of learning the nonlinear time evolution of a probability distribution. Capability of most of the conventional time series models is limited when the underlying dynamics is nonlinear, multi-scale or when there is no prior knowledge at all on the system dynamics. We propose DE-RNN (Density Estimation Recurrent Neural Network) to learn the probability density function (PDF) of a stochastic process with an underlying nonlinear dynamics and compute the time evolution of the PDF for a probabilistic forecast. A Recurrent Neural Network (RNN)-based model is employed to learn a nonlinear operator for the temporal evolution of the stochastic process. We use a softmax layer for a numerical discretization of a smooth PDF, which transforms a function approximation problem to a classification task. A regularized cross-entropy method is introduced to impose a smoothness condition on the estimated probability distribution. A Monte Carlo procedure to compute the temporal evolution of the distribution for a multiple-step forecast is presented. It is shown that the proposed algorithm can learn the nonlinear multi-scale dynamics from the noisy observations and provides an effective tool to forecast time evolution of the underlying probability distribution. Evaluation of the algorithm on three synthetic and two real data sets shows advantage over the compared baselines, and a potential value to a wide range of problems in physics and engineering.