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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

PySIDT: Subgraph Isomorphic Decision Trees for Molecular Property Prediction

Accurate molecular property prediction is important across all fields of chemistry. Deep neural networks (DNNs) have become increasingly popular due to their ability to train automatically, avoiding the incredibly tedious process of constructing and extending traditional property estimation schemes. However, DNNs require large amounts of training data, are challenging to interpret, require large amounts of memory to load even during inference, and have severe difficulties incorporating qualitative chemical knowledge, which are often desired for molecular property prediction tasks. Here, in this study, we present PySIDT (https://github.com/zadorlab/PySIDT), a software for training and running inference on Subgraph Isomorphic Decision Trees (SIDTs). SIDTs are graph-based decision trees made of nodes associated with molecular substructures. Inference is done by descending target molecular structures down the decision tree to nodes with matching subgraph isomorphic substructures and making predictions based on the final (most specific) nodes matched. SIDTs scale down well to dataset sizes much smaller than is feasible for DNNs. As trees of molecular substructures, SIDTs are inherently readable and easy to visualize, making them easy to analyze. They are also straightforward to extend and retrain, facilitate uncertainty estimation, and enable easy integration of expert knowledge. We demonstrate the SIDT approach discussing its application to a diverse range of molecular prediction tasks: rate coefficient estimation, diffusion coefficient estimation, thermochemistry estimation, transition state bond stretch prediction, p K a prediction, stability of molecular structures, stability of surface structures, and prediction of surface lateral interaction energetics. Additionally, we demonstrate the power of the SIDT algorithms in two direct learning curve vanilla comparisons with the popular DNN-based software Chemprop and the popular gradient boosted trees-based software XGBoost on enthalpy of formation and rate coefficient prediction tasks. In particular, in the enthalpy of formation case, vanilla PySIDT is able to outperform vanilla Chemprop and XGBoost across the full range of training/validation set sizes out to 11,560 data points.

Johnson, Matthew Sean [Sandia National Laboratorie↗
DOE OSTI2021

Towards data-driven constitutive modelling for granular materials via micromechanics-informed deep learning

The analytical description of path-dependent elastic-plastic responses of a granular system is highly complicated because of continuously evolving microstructures and strain localisation within the system undergoing deformation. This study offers an alternative to the current analytical paradigm by developing micromechanics-informed machine-learning based constitutive modelling approaches for granular materials. A set of critical variables associated with the constitutive behaviour of granular materials are identified through an incremental stress-strain relationship analysis. Depending on the strategy to exploit the priori micromechanical knowledge, three different training strategies are explored. The first model uses only the measurable external variables to make stress predictions; the second model utilises a directed graph to link all the external strain sequences and internal microstructural evolution variables into a single prediction model comprised of a series of sub-mappings, and the third model explicitly integrates the physically important non-temporal properties with external strain paths into training through an enhanced Gated Recurrent Unit (GRU). These three models show satisfactory agreement with unseen test specimens based on multi-directional loading cases. The basic features and potential applications of each model are explained. Lastly, the key factors for constitutive training and limitations of the current work are also discussed in detail.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Global normal spectral irradiance in Albuquerque: a one-year open dataset for PV research

This report describes the creation process and final content of a spectral irradiance dataset for Albuquerque NM. The spectral irradiance measurements were made using a dual-axis tracker; therefore, they represent global normal irradiance. The dataset combines spectroradiometer and weather measurements from a two-year period into a continuous calendar year. The data files are accompanied by extensive metadata as well as example calculations and graphs to demonstrate the potential uses of this database.

14 SOLAR ENERGY↗
DOE OSTI2023

An uncertainty-aware strategy for plasma mechanism reduction with directed weighted graphs

In this work, we present a framework for the analysis and reduction of plasma mechanisms by means of weighted directed graphs, in which reactions and species are both treated as nodes. The methodology consists of two distinct analyses. The first, which is qualitative, relies on graph spatializations via force-directed algorithms to discover the predominant global patterns in the chemical model. The second ranks the reactions based on their shortest paths' lengths from/to the species of interest and their relative contributions to the power balance. Further, this quantitative investigation enables a strategy for mechanism reduction that is fully automatized, as it does not require any expert knowledge, highly effective, as it generates reduced mechanisms that are highly accurate while relying on a small number of processes, and easily interpretable, as the algorithm justifies the importance of the retained reactions by outputting their related chemical pathways. Additionally, the work proposes a methodology extension that employs ensembles of graphs to improve the robustness of the reduced mechanism to reaction parameter uncertainties. The approach, here tested for steady-state predictions of a plasma system characterizing negative hydrogen ion sources, is general and can be used in a wide variety of applications outside the particular nuclear fusion context demonstrated in this work.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Trust: Triangle Counting Reloaded on GPUs

Triangle counting is a building block for a wide range of graph applications. Here, traditional wisdom suggests that i) hashing is not suitable for triangle counting, ii) edge-centric triangle counting beats vertex-centric design, and iii) communication-free and workload balanced graph partitioning is a grand challenge for triangle counting. On the contrary, we advocate that i) hashing can help the key operations for scalable triangle counting on Graphics Processing Units (GPUs), i.e., list intersection and graph partitioning, ii) vertex-centric option reduces both hash table construction cost and memory consumption, which is limited on GPUs. In addition, iii) we exploit graph and workload collaborative, and hash-based 2D partitioning to scale vertex-centric triangle counting over 1,000 GPUs with sustained scalability. In this work, we present TRUST, which performs triangle counting with the hash operation and vertex-centric paradigm. To the best of our knowledge, TRUST is the first work that achieves over one trillion Traversed Edges Per Second (TEPS) rate for triangle counting.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Scalable Pattern Matching in Metadata Graphs via Constraint Checking

Pattern matching is a fundamental tool for answering complex graph queries. Unfortunately, existing solutions have limited capabilities: They do not scale to process large graphs and/or support only a restricted set of search templates or usage scenarios. Moreover, the algorithms at the core of the existing techniques are not suitable for today’s graph processing infrastructures relying on horizontal scalability and shared-nothing clusters, as most of these algorithms are inherently sequential and difficult to parallelize. In this article we present an algorithmic pipeline that bases pattern matching on constraint checking. The key intuition is that each vertex and edge participating in a match has to meet a set of constraints implicitly specified by the search template. These constraints can be verified independently and typically are less expensive to compute than searching the full template. The pipeline we propose generates these constraints and iterates over them to eliminate all the vertices and edges that do not participate in any match, thus reducing the background graph to a subgraph that is the union of all template matches—the complete set of all vertices and edges that participate in at least one match. Additional analysis can be performed on this annotated, reduced graph, such as full match enumeration, match counting, or computing vertex/edge centrality. Furthermore, a vertex-centric formulation for constraint checking algorithms exists, and this makes it possible to harness existing high-performance, vertex-centric graph processing frameworks. This technique (i) enables highly scalable pattern matching in metadata (labeled) graphs; (ii) supports arbitrary patterns with 100% precision; (iii) enables tradeoffs between precision and time-to-solution, while always selects all vertices and edges that participate in matches, thus offering 100% recall; and (iv) supports a set of popular data analytics scenarios. We implement our approach on top of HavoqGT, an open-source asynchronous graph processing framework, and demonstrate its advantages through strong and weak scaling experiments on massive scale real-world (up to 257 billion edges) and synthetic (up to 4.4 trillion edges) labeled graphs, respectively, and at scales (1,024 nodes / 36,864 cores), orders of magnitude larger than used in the past for similar problems. This article serves two purposes: First, it synthesises the knowledge accumulated during a long-term project. Second, it presents new system features, usage scenarios, optimizations, and comparisons with related work that strengthen the confidence that pattern matching based on iterative pruning via constraint checking is an effective and scalable approach in practice. The new contributions include the following: (i) We demonstrate the ability of the constraint checking approach to efficiently support two additional search scenarios that often emerge in practice, interactive incremental search and exploratory search. (ii) We empirically compare our solution with two additional state-of-the-art systems, Arabsque and TriAD. (iii) We show the ability of our solution to accommodate a more diverse range of datasets with varying properties, e.g., scale, skewness, label distribution, and match frequency. (iv) We introduce or extend a number of system features (e.g., work aggregation, load balancing, and the ability to cap the generated traffic) and design optimizations and demonstrate their advantages with respect to improving performance and scalability. (v) We present bottleneck analysis and insights into artifacts that influence performance. (vi) We present a theoretical complexity argument that motivates the performance gains we observe.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Using spatio-temporal graph neural networks to estimate fleet-wide photovoltaic performance degradation patterns

Accurate estimation of photovoltaic (PV) system performance is crucial for determining its feasibility as a power generation technology and financial asset. PV-based energy solutions offer a viable alternative to traditional energy resources due to their superior Levelized Cost of Energy (LCOE). A significant challenge in assessing the LCOE of PV systems lies in understanding the Performance Loss Rate (PLR) for large fleets of PV systems. Estimating the PLR of PV systems becomes increasingly important in the rapidly growing PV industry. Precise PLR estimation benefits PV users by providing real-time monitoring of PV module performance, while explainable PLR estimation assists PV manufacturers in studying and enhancing the performance of their products. However, traditional PLR estimation methods based on statistical models have notable drawbacks. Firstly, they require user knowledge and decision-making. Secondly, they fail to leverage spatial coherence for fleet-level analysis. Additionally, these methods inherently assume the linearity of degradation, which is not representative of real world degradation. To overcome these challenges, we propose a novel graph deep learning-based decomposition method called the Spatio-Temporal Graph Neural Network for fleet-level PLR estimation (PV-stGNN-PLR). PV-stGNN-PLR decomposes the power timeseries data into aging and fluctuation components, utilizing the aging component to estimate PLR. PV-stGNN-PLR exploits spatial and temporal coherence to derive PLR estimation for all systems in a fleet and imposes flatness and smoothness regularization in loss function to ensure the successful disentanglement between aging and fluctuation. We have evaluated PV-stGNN-PLR on three simulated PV datasets consisting of 100 inverters from 5 sites. Experimental results show that PV-stGNN-PLR obtains a reduction of 33.9% and 35.1% on average in Mean Absolute Percent Error (MAPE) and Euclidean Distance (ED) in PLR degradation pattern estimation compared to the state-of-the-art PLR estimation methods.

14 SOLAR ENERGY↗
DOE OSTI2026

Improving the Performance of NEML2 with Modern Graph Compilation Backends

NEML2 vectorizes constitutive-model evaluation for large-scale multiphysics simulation, using PyTorch as its tensor backend so that a batch of material-point updates runs on CPU or GPU through a single implementation. In the two prior reports in this series it was a C++-native library, deployed through TorchScript tracing and just-in-time (JIT) compilation; it has since been rewritten from the ground up into a Python-native library deployed through Ahead-of-Time Inductor (AOTInductor), a modern PyTorch graph-compilation backend. The rewrite is driven by a persistent tension, not a language preference: NEML2 composes constitutive models at runtime from a registry of small, independently-authored pieces, and that flexibility is difficult to reconcile with the compile-time knowledge an efficient GPU kernel needs. This report documents the rewrite and the investment that accompanied it: the AOTInductor export pipeline that turns a Python-authored model into a portable, Python-free compiled artifact loadable from pure C++; the eager and compiled runtimes and the new implicit solver layer built on them; a head-to-head benchmark of legacy JIT against AOTInductor; the physics-model catalog and its worked examples; the developer tooling; and the corresponding overhaul of MOOSE’s NEML2 integration that lets MOOSE consume it. A central objective is to examine whether modern PyTorch graph-compilation backends are effective for MOOSE GPU integration. The benchmark answers directly: AOTInductor outperforms legacy JIT on every GPU scenario measured, by 1.0–4.5×. Modern graph-compilation backends are effective for MOOSE GPU integration, and AOTInductor specifically – not compilation in the abstract – is why.

Hu, Gary (Tianchen) [Argonne National Laboratory (↗
DOE OSTI2025

Integrated Framework of Multisource Data Fusion for Outage Location in Looped Distribution Systems

Accurate outage location is essential for expediting post-outage power restoration, minimizing outage duration, and enhancing the resilience of distribution networks. With the advent of advanced metering infrastructure, data-driven outage location methods have significantly advanced beyond traditional approaches that rely on manual inspections. However, existing methods still face critical challenges, like reliance on single-source data, limited ability to handle partially observable systems or difficulties with loop networks. To the best of our knowledge, no single approach has comprehensively addressed all of these challenges at once. To this end, this paper proposes a comprehensive multisource data fusion framework for outage locations via probabilistic graph networks. The framework consists of three key phases. First, a novel method for reconstituting distribution networks with loops is developed, transforming looped networks into multiple radial subnetworks that retain all outage causalities of the original network. Second, Bayesian network (BN) models are established for each subnetwork, integrating multiple data sources and network structures. Finally, a joint Gibbs sampling mechanism, featuring forward and backward information flow, is designed to merge data from separate BN models and maximize the utilization of limited evidence, ensuring accurate outage location identification. In conclusion, the framework was validated on two modified public test systems, and comparative studies confirmed its effectiveness.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

HDBind: encoding of molecular structure with hyperdimensional binary representations

Traditional methods for identifying “hit” molecules from a large collection of potential drug-like candidates rely on biophysical theory to compute approximations to the Gibbs free energy of the binding interaction between the drug and its protein target. These approaches have a significant limitation in that they require exceptional computing capabilities for even relatively small collections of molecules. Increasingly large and complex state-of-the-art deep learning approaches have gained popularity with the promise to improve the productivity of drug design, notorious for its numerous failures. However, as deep learning models increase in their size and complexity, their acceleration at the hardware level becomes more challenging. Hyperdimensional Computing (HDC) has recently gained attention in the computer hardware community due to its algorithmic simplicity relative to deep learning approaches. The HDC learning paradigm, which represents data with high-dimension binary vectors, allows the use of low-precision binary vector arithmetic to create models of the data that can be learned without the need for the gradient-based optimization required in many conventional machine learning and deep learning methods. This algorithmic simplicity allows for acceleration in hardware that has been previously demonstrated in a range of application areas (computer vision, bioinformatics, mass spectrometery, remote sensing, edge devices, etc.). To the best of our knowledge, our work is the first to consider HDC for the task of fast and efficient screening of modern drug-like compound libraries. We also propose the first HDC graph-based encoding methods for molecular data, demonstrating consistent and substantial improvement over previous work. We compare our approaches to alternative approaches on the well-studied MoleculeNet dataset and the recently proposed LIT-PCBA dataset derived from high quality PubChem assays. We demonstrate our methods on multiple target hardware platforms, including Graphics Processing Units (GPUs) and Field Programmable Gate Arrays (FPGAs), showing at least an order of magnitude improvement in energy efficiency versus even our smallest neural network baseline model with a single hidden layer. Our work thus motivates further investigation into molecular representation learning to develop ultra-efficient pre-screening tools. We make our code publicly available at https://github.com/LLNL/hdbind.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2026

py-boomer v0.1.0

Py-BOOMER (Python Bayesian OWL Ontology MErgER in Python) is a probabilistic reasoning system for knowledge representation and ontological reasoning with uncertainty. Itnables reasoning over probabilistic facts and taxonomic relationships, finding the most likely consistent interpretation of potentially conflicting assertions. It uses a combination of graph-based reasoning and Bayesian probabilistic inference. Key features: Represent probabilistic ontological statements Reason over class subsumption hierarchies Evaluate class equivalence relationships Detect and resolve logical inconsistencies Calculate posterior probabilities for each assertion

Mungall, Chris [Lawrence Berkeley National Laborat↗
DOE OSTI2014

On Parallel Push-Relabel based Algorithms for Bipartite Maximum Matching

We study multithreaded push-relabel based algorithms for computing maximum cardinality matching in bipartite graphs. Matching is a fundamental combinatorial (graph) problem with applications in a wide variety of problems in science and engineering. We are motivated by its use in the context of sparse linear solvers for computing maximum transversal of a matrix. We implement and test our algorithms on several multi-socket multicore systems and compare their performance to state-of-the-art augmenting path-based serial and parallel algorithms using a testset comprised of a wide range of real-world instances. Building on several heuristics for enhancing performance, we demonstrate good scaling for the parallel push-relabel algorithm. We show that it is comparable to the best augmenting path-based algorithms for bipartite matching. To the best of our knowledge, this is the first extensive study of multithreaded push-relabel based algorithms. In addition to a direct impact on the applications using matching, the proposed algorithmic techniques can be extended to preflow-push based algorithms for computing maximum flow in graphs.

Push-relabel, matching, bipartite↗
DOE OSTI2019

Graph Generation with a Focusing Lexicon

Reference datasets are an essential part of developing effective data exploration tools. Many reference datasets are synthetic in nature, seeding real-world properties into generation routines. Others are laboriously curated from original source materials. This is a time-consuming process that requires significant domain expertise all along the process. This paper presents a third path: deriving a network from real-world source materials using automated tools. By properly grounding the tools in a small collection of curated documents, a larger corpus be automatically processed with greater confidence. The end system leverages human knowledge (in the initial document curation), but with much less time commitment enabled by automation. Our ultimate goal is to produce reference datasets, derived from real-world data that capture aspects salient to a specific activity in the context of all of the activities in the data-set. Our reference datasets take the form of graphs.

Chandra Shekar, Mayanka↗
DOE OSTI2017

Technical Evaluation Report for the NDMAS AGR-5/6/7 TRISO Irradiation Monitoring System

The Nuclear Data Management and Analysis System (NDMAS) manages irradiation experiment data collected under the Advanced Gas Reactor (AGR) Fuel Development and Qualification program within the Advanced Reactor Technologies Technology Development Office. The AGR-5/6/7 irradiation experiment combines the fifth, sixth, and seventh planned irradiation campaigns of the U.S. Department of Energy AGR Fuel Development and Qualification Program. These combined irradiation tests will consist of fuel qualification tests (AGR-5/6) and a margin test (AGR-7) for the industrially produced uranium oxycarbide (UCO) tristructural isotropic (TRISO) coated particle fuel developed by the AGR program. The objective of the AGR-5/6/7 irradiation experiment is to establish the performance of UCO TRISO fuel fabricated using engineering-scale equipment in a production environment under and beyond normal operating conditions (PNL-5245). Experimental irradiation measurements for this experiment (i.e., thermocouple temperature, gas flow rate, fission product release rate, gross gamma count, and Advanced Test Reactor (ATR) operational parameters) are captured automatically to the NDMAS database. Data derived from numerical simulation or other mathematical analyses (i.e., fast fluence, burnup, temperature, and fission product release to birth ratio) are captured to the database when provided by the researchers. The NDMAS database is a relational database accessed via structured query language and managed with Microsoft SQL Server. While select users can directly access the database, that is not the most convenient way to interact with the data. The NDMAS team constructs numerous tables that combine related datasets, and delivers those data to researchers via graphs and spreadsheets on a Microsoft SharePoint-based website. For the AGR-5/6/7 experimental data, a new subsite has been developed, along with a set of web pages. These pages will be used by members of fuel irradiation team for monitoring irradiation conditions according to specification (SPC-1749), by the physics modelers to access necessary model inputs, and by the nuclear engineering community to view and access data such as fuel temperatures, fission product release from capsules, and ATR operational parameters. The deliverable represented in this document is a SharePoint website containing tables of the AGR-5/6/7 measured and calculated data, graphs displaying the data, and links to source documents and references.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2022

Projecting the Thermal Response in a HTGR-Type System during Conduction Cooldown Using Graph-Laplacian Based Machine Learning

Accurate prediction of an off-normal event in a nuclear reactor is dependent upon the availability of sensory data, reactor core physical condition, and understanding of the underlying phenomenon. This work presents a method to project the data from some discrete sensory locations to the overall reactor domain during conduction cooldown scenarios similar to High Temperature Gas-cooled Reactors (HTGRs). The existing models for conductive cooldown in a heterogeneous multi-body system, such as an assembly of prismatic blocks or pebble beds relies on knowledge of the thermal contact conductance, requiring significant knowledge of local thermal contacts and heat transport possibilities across those contacts. With a priori knowledge of bulk geometry features and some discrete sensors, a machine learning approach was devised. The presented work uses an experimental facility to mimic conduction cooldown with an assembly of 68 cylindrical rods initially heated to 1200 K. High-fidelity temperature data were collected using an infrared (IR) camera to provide training data to the model and validate the predicted temperature data. The machine learning approach used here first converts the macroscopic bulk geometry information into Graph-Laplacian, and then uses the eigenvectors of the Graph-Laplacian to develop Kernel functions. Support vector regression (SVR) was implemented on the obtained Kernels and used to predict the thermal response in a packed rod assembly during a conduction cooldown experiment. The usage of SVR modeling differs from most models today because of its representation of thermal coupling between rods in the core. When trained with thermographic data, the average normalized error is less than 2% over 400 s, during which temperatures of the assembly have dropped by more than 500 K. The rod temperature prediction performance was significantly better for rods in the interior of the assembly compared to those near the exterior, likely due to the model simplification of the surroundings.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2023

A unified understanding of minimum lattice thermal conductivity

Here, we propose a first-principles model of minimum lattice thermal conductivity ($κ^{min}_L$) based on a unified theoretical treatment of thermal transport in crystals and glasses. We apply this model to thousands of inorganic compounds and find a universal behavior of $κ^{min}_L$ in crystals in the high-temperature limit: The isotropically averaged $κ^{min}_L$ is independent of structural complexity and bounded within a range from ~0.1 to ~2.6 W/(m K), in striking contrast to the conventional phonon gas model which predicts no lower bound. We unveil the underlying physics by showing that for a given parent compound, $κ^{min}_L$ is bounded from below by a value that is approximately insensitive to disorder, but the relative importance of different heat transport channels (phonon gas versus diffuson) depends strongly on the degree of disorder. Moreover, we propose that the diffuson-dominated $κ^{min}_L$ in complex and disordered compounds might be effectively approximated by the phonon gas model for an ordered compound by averaging out disorder and applying phonon unfolding. With these insights, we further bridge the knowledge gap between our model and the well-known Cahill–Watson–Pohl (CWP) model, rationalizing the successes and limitations of the CWP model in the absence of heat transfer mediated by diffusons. Finally, we construct graph network and random forest machine learning models to extend our predictions to all compounds within the Inorganic Crystal Structure Database (ICSD), which were validated against thermoelectric materials possessing experimentally measured ultralow κ L . Our work offers a unified understanding of $κ^{min}_L$, which can guide the rational engineering of materials to achieve .

42 ENGINEERING↗
DOE OSTI2025

Meta-Learning Enhanced Physics-Informed Graph Attention Convolutional Network for Distribution Power System State Estimation

Promptly perceiving distribution system states is challenged by frequent topology changes and uncertain power injections. To address these issues, a Meta-learning enhanced physics-informed graph attention convolutional network (Meta-PIGACN) model is proposed to handle topological variability in distribution system state estimation (DSSE). Specifically, physics information is integrated into the graph convolutional network, enabling a physics-informed edge-weighting process that incorporates physical information to control the aggregation of neighboring nodes. Besides, the graph attention mechanism automatically adjusts the importance of different neighboring nodes, allowing the capture and preservation of inherent system features across varying topologies, thereby improving state estimation accuracy. Furthermore, meta-learning is proposed to acquire empirical knowledge across multiple topologies so that the model can rapidly adapt to new configurations through iterative gradient descent updates even in large-scale systems. In conclusion, the simulation results based on the 33/118/1746-node distribution systems show the high accuracy and efficiency of the proposed model.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Learning Distribution Grid Topologies: A Tutorial

Unveiling feeder topologies from data is of paramount importance to advance situational awareness and proper utilization of smart resources in power distribution grids. This tutorial summarizes, contrasts, and establishes useful links between recent works on topology identification and detection schemes that have been proposed for power distribution grids. The primary focus is to highlight methods that overcome the limited availability of measurement devices in distribution grids, while enhancing topology estimates using conservation laws of power-flow physics and structural properties of feeders. Grid data from phasor measurement units or smart meters can be collected either passively in the traditional way, or actively, upon actuating grid resources and measuring the feeder's voltage response. Analytical claims on feeder identifiability and detectability are reviewed under disparate meter placement scenarios. Such topology learning claims can be attained exactly or approximately so via algorithmic solutions with various levels of computational complexity, ranging from least-squares fits to convex optimization problems, and from polynomial-time searches over graphs to mixed-integer programs. Although the emphasis is on radial single-phase feeders, extensions to meshed and/or multiphase circuits are sometimes possible and discussed. Here this tutorial aspires to provide researchers and engineers with knowledge of the current state-of-the-art in tractable distribution grid learning and insights into future directions of work.

24 POWER TRANSMISSION AND DISTRIBUTION↗