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Nowadays, the data collected in physical/engineering systems allows various machine learning methods to conduct system monitoring and control, when the physical knowledge on the system edge is limited and challenging to recover completely. Solving such problems typically requires identifying forward system mapping rules, from system states to the output measurements. However, the forward system identification based on digital twin can hardly provide complete monitoring functions, such as state estimation, e.g., to infer the states from measurements. While one can directly learn the inverse mapping rule, it is more desirable to re-utilize the forward digital twin since it is relatively easy to embed physical law there to regularize the inverse process and avoid overfitting. For this purpose, this paper proposes an invertible learning structure based on designing parallel paths in structural neural networks with basis functionals and embedding virtual storage variables for information preservation. For such a two-way digital twin modeling, there is an additional challenge of multiple solutions for system inverse, which contradict the reality of one feasible solution for the current system. To avoid ambiguous inverse, the proposed model maximizes the physical likelihood to contract the original solution space, leading to the unique system operation status of interest. We validate the proposed method on various physical system monitoring tasks and scenarios, such as inverse kinematics problems, power system state estimation, etc. Furthermore, by building a perfect match of a forward-inverse pair, the proposed method obtains accurate and computation-efficient inverse predictions, given observations. Finally, the forward physical interpretation and small prediction errors guarantee the explainability of the invertible structure, compared to standard learning methods. 

Learning the underlying equation from data is a fundamental problem in many disciplines. Recent advances rely on Neural Networks (NNs) but do not provide theoretical guarantees in obtaining the exact equations owing to the non-convexity of NNs. In this paper, we propose Convex Neural Symbolic Learning (CoNSoLe) to seek convexity under mild conditions. The main idea is to decompose the recovering process into two steps and convexify each step. In the first step of searching for right symbols, we convexify the deep Q-learning. The key is to maintain double convexity for both the negative Q-function and the negative reward function in each iteration, leading to provable convexity of the negative optimal Q function to learn the true symbol connections. Conditioned on the exact searching result, we construct a Locally Convex equation Learning (LoCaL) neural network to convexify the estimation of symbol coefficients. With such a design, we quantify a large region with strict convexity in the loss surface of LoCaL for commonly used physical functions. Finally, we demonstrate the superior performance of the CoNSoLe framework over the state-of-the-art on a diverse set of datasets. 

The growing integration of distributed energy resources (DERs) in distribution grids raises various reliability issues due to DER's uncertain and complex behaviors. With large-scale DER penetration in distribution grids, traditional outage detection methods, which rely on customers report and smart meters' “last gasp” signals, will have poor performance, because renewable generators and storage and the mesh structure in urban distribution grids can continue supplying power after line outages. To address these challenges, we propose a data-driven outage monitoring approach based on the stochastic time series analysis with a theoretical guarantee. Specifically, we prove via power flow analysis that dependency of time-series voltage measurements exhibits significant statistical changes after line outages. This makes the theory on optimal change-point detection suitable to identify line outages. However, existing change point detection methods require post-outage voltage distribution, which are unknown in distribution systems. Therefore, we design a maximum likelihood estimator to directly learn distribution pa-rameters from voltage data. We prove the estimated parameters-based detection also achieves optimal performance, making it extremely useful for fast distribution grid outage identifications. Furthermore, since smart meters have been widely installed in distribution grids and advanced infrastructure (e.g., PMU) has not widely been available, our approach only requires voltage magnitude for quick outage identification. Simulation results show highly accurate outage identification in eight distribution grids with 17 configurations with and without DERs using smart meter data.