Implementing accurate Distribution System State Estimation (DSSE) faces several challenges, among which the lack of observability and the high density of the distribution system. While data-driven alternatives based on Machine Learning models could be a choice, they suffer in DSSE because of the lack of labeled data. In fact, measurements in the distribution system are often noisy, corrupted, and unavailable. To address these issues, we propose the Deep Statistical Solver for Distribution System State Estimation (DSS$^2$), a deep learning model based on graph neural networks (GNNs) that accounts for the network structure of the distribution system and for the physical governing power flow equations. DSS$^2$ leverages hypergraphs to represent the heterogeneous components of the distribution systems and updates their latent representations via a node-centric message-passing scheme. A weakly supervised learning approach is put forth to train the DSS$^2$ in a learning-to-optimize fashion w.r.t. the Weighted Least Squares loss with noisy measurements and pseudomeasurements. By enforcing the GNN output into the power flow equations and the latter into the loss function, we force the DSS$^2$ to respect the physics of the distribution system. This strategy enables learning from noisy measurements, acting as an implicit denoiser, and alleviating the need for ideal labeled data. Extensive experiments with case studies on the IEEE 14-bus, 70-bus, and 179-bus networks showed the DSS$^2$ outperforms by a margin the conventional Weighted Least Squares algorithm in accuracy, convergence, and computational time, while being more robust to noisy, erroneous, and missing measurements. The DSS$^2$ achieves a competing, yet lower, performance compared with the supervised models that rely on the unrealistic assumption of having all the true labels.
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Given univariate random variables $Y_1, \ldots, Y_n$ with the $\text{Uniform}(\theta_0 - 1, \theta_0 + 1)$ distribution, the sample midrange $\frac{Y_{(n)}+Y_{(1)}}{2}$ is the MLE for $\theta_0$ and estimates $\theta_0$ with error of order $1/n$, which is much smaller compared with the $1/\sqrt{n}$ error rate of the usual sample mean estimator. However, the sample midrange performs poorly when the data has say the Gaussian $N(\theta_0, 1)$ distribution, with an error rate of $1/\sqrt{\log n}$. In this paper, we propose an estimator of the location $\theta_0$ with a rate of convergence that can, in many settings, adapt to the underlying distribution which we assume to be symmetric around $\theta_0$ but is otherwise unknown. When the underlying distribution is compactly supported, we show that our estimator attains a rate of convergence of $n^{-\frac{1}{\alpha}}$ up to polylog factors, where the rate parameter $\alpha$ can take on any value in $(0, 2]$ and depends on the moments of the underlying distribution. Our estimator is formed by the $\ell^\gamma$-center of the data, for a $\gamma\geq2$ chosen in a data-driven way -- by minimizing a criterion motivated by the asymptotic variance. Our approach can be directly applied to the regression setting where $\theta_0$ is a function of observed features and motivates the use of $\ell^\gamma$ loss function for $\gamma > 2$ in certain settings.
Out-of-distribution detection is one of the most critical issue in the deployment of machine learning. The data analyst must assure that data in operation should be compliant with the training phase as well as understand if the environment has changed in a way that autonomous decisions would not be safe anymore. The method of the paper is based on eXplainable Artificial Intelligence (XAI); it takes into account different metrics to identify any resemblance between in-distribution and out of, as seen by the XAI model. The approach is non-parametric and distributional assumption free. The validation over complex scenarios (predictive maintenance, vehicle platooning, covert channels in cybersecurity) corroborates both precision in detection and evaluation of training-operation conditions proximity.
Mathematical models are essential for understanding and making predictions about systems arising in nature and engineering. Yet, mathematical models are a simplification of true phenomena, thus making predictions subject to uncertainty. Hence, the ability to quantify uncertainties is essential to any modelling framework, enabling the user to assess the importance of certain parameters on quantities of interest and have control over the quality of the model output by providing a rigorous understanding of uncertainty. Peridynamic models are a particular class of mathematical models that have proven to be remarkably accurate and robust for a large class of material failure problems. However, the high computational expense of peridynamic models remains a major limitation, hindering outer-loop applications that require a large number of simulations, for example, uncertainty quantification. This contribution provides a framework to make such computations feasible. By employing a Multilevel Monte Carlo (MLMC) framework, where the majority of simulations are performed using a coarse mesh, and performing relatively few simulations using a fine mesh, a significant reduction in computational cost can be realised, and statistics of structural failure can be estimated. The results show a speed-up factor of 16x over a standard Monte Carlo estimator, enabling the forward propagation of uncertain parameters in a computationally expensive peridynamic model. Furthermore, the multilevel method provides an estimate of both the discretisation error and sampling error, thus improving the confidence in numerical predictions. The performance of the approach is demonstrated through an examination of the statistical size effect in quasi-brittle materials.
This paper is dedicated to achieving scalable relative state estimation using inter-robot Euclidean distance measurements. We consider equipping robots with distance sensors and focus on the optimization problem underlying relative state estimation in this setup. We reveal the commonality between this problem and the coordinates realization problem of a sensor network. Based on this insight, we propose an effective unconstrained optimization model to infer the relative states among robots. To work on this model in a distributed manner, we propose an efficient and scalable optimization algorithm with the classical block coordinate descent method as its backbone. This algorithm exactly solves each block update subproblem with a closed-form solution while ensuring convergence. Our results pave the way for distance measurements-based relative state estimation in large-scale multi-robot systems.
Noise Contrastive Estimation (NCE) is a popular approach for learning probability density functions parameterized up to a constant of proportionality. The main idea is to design a classification problem for distinguishing training data from samples from an easy-to-sample noise distribution $q$, in a manner that avoids having to calculate a partition function. It is well-known that the choice of $q$ can severely impact the computational and statistical efficiency of NCE. In practice, a common choice for $q$ is a Gaussian which matches the mean and covariance of the data. In this paper, we show that such a choice can result in an exponentially bad (in the ambient dimension) conditioning of the Hessian of the loss, even for very simple data distributions. As a consequence, both the statistical and algorithmic complexity for such a choice of $q$ will be problematic in practice, suggesting that more complex noise distributions are essential to the success of NCE.
Sampling from a target measure whose density is only known up to a normalization constant is a fundamental problem in computational statistics and machine learning. In this paper, we present a new optimization-based method for sampling called mollified interaction energy descent (MIED). MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs). These energies rely on mollifier functions -- smooth approximations of the Dirac delta originated from PDE theory. We show that as the mollifier approaches the Dirac delta, the MIE converges to the chi-square divergence with respect to the target measure and the gradient flow of the MIE agrees with that of the chi-square divergence. Optimizing this energy with proper discretization yields a practical first-order particle-based algorithm for sampling in both unconstrained and constrained domains. We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD, while for constrained sampling problems our method readily incorporates constrained optimization techniques to handle more flexible constraints with strong performance compared to alternatives.
Legged robot navigation in unstructured and slippery terrains depends heavily on the ability to accurately identify the quality of contact between the robot's feet and the ground. Contact state estimation is regarded as a challenging problem and is typically addressed by exploiting force measurements, joint encoders and/or robot kinematics and dynamics. In contrast to most state of the art approaches, the current work introduces a novel probabilistic method for estimating the contact state based solely on proprioceptive sensing, as it is readily available by Inertial Measurement Units (IMUs) mounted on the robot's end effectors. Capitalizing on the uncertainty of IMU measurements, our method estimates the probability of stable contact. This is accomplished by approximating the multimodal probability density function over a batch of data points for each axis of the IMU with Kernel Density Estimation. The proposed method has been extensively assessed against both real and simulated scenarios on bipedal and quadrupedal robotic platforms such as ATLAS, TALOS and Unitree's GO1.
We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of $C^{\infty}$ bump functions of varying support sizes. We demonstrate that WENDy is a highly robust and efficient method for parameter inference in differential equations. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. We illustrate the method and its performance in some common population and neuroscience models, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at (//github.com/MathBioCU/WENDy).
This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.
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