Space-Time Bridge-Diffusion

In this study, we introduce a novel method for generating new synthetic samples that are independent and identically distributed (i.i.d.) from high-dimensional real-valued probability distributions, as defined implicitly by a set of Ground Truth (GT) samples. Central to our method is the integration of space-time mixing strategies that extend across temporal and spatial dimensions. Our methodology is underpinned by three interrelated stochastic processes designed to enable optimal transport from an easily tractable initial probability distribution to the target distribution represented by the GT samples: (a) linear processes incorporating space-time mixing that yield Gaussian conditional probability densities, (b) their bridge-diffusion analogs that are conditioned to the initial and final state vectors, and (c) nonlinear stochastic processes refined through score-matching techniques. The crux of our training regime involves fine-tuning the nonlinear model, and potentially the linear models - to align closely with the GT data. We validate the efficacy of our space-time diffusion approach with numerical experiments, laying the groundwork for more extensive future theory and experiments to fully authenticate the method, particularly providing a more efficient (possibly simulation-free) inference.

U-Turn Diffusion

We present a comprehensive examination of score-based diffusion models of AI for generating synthetic images. These models hinge upon a dynamic auxiliary time mechanism driven by stochastic differential equations, wherein the score function is acquired from input images. Our investigation unveils a criterion for evaluating efficiency of the score-based diffusion models: the power of the generative process depends on the ability to de-construct fast correlations during the reverse/de-noising phase. To improve the quality of the produced synthetic images, we introduce an approach coined "U-Turn Diffusion". The U-Turn Diffusion technique starts with the standard forward diffusion process, albeit with a condensed duration compared to conventional settings. Subsequently, we execute the standard reverse dynamics, initialized with the concluding configuration from the forward process. This U-Turn Diffusion procedure, combining forward, U-turn, and reverse processes, creates a synthetic image approximating an independent and identically distributed (i.i.d.) sample from the probability distribution implicitly described via input samples. To analyze relevant time scales we employ various analytical tools, including auto-correlation analysis, weighted norm of the score-function analysis, and Kolmogorov-Smirnov Gaussianity test. The tools guide us to establishing that the Kernel Intersection Distance, a metric comparing the quality of synthetic samples with real data samples, is minimized at the optimal U-turn time.

A Physics-Informed Machine Learning for Electricity Markets: A NYISO Case Study

This paper addresses the challenge of efficiently solving the optimal power flow problem in real-time electricity markets. The proposed solution, named Physics-Informed Market-Aware Active Set learning OPF (PIMA-AS-OPF), leverages physical constraints and market properties to ensure physical and economic feasibility of market-clearing outcomes. Specifically, PIMA-AS-OPF employs the active set learning technique and expands its capabilities to account for curtailment in load or renewable power generation, which is a common challenge in real-world power systems. The core of PIMA-AS-OPF is a fully-connected neural network that takes the net load and the system topology as input. The outputs of this neural network include active constraints such as saturated generators and transmission lines, as well as non-zero load shedding and wind curtailments. These outputs allow for reducing the original market-clearing optimization to a system of linear equations, which can be solved efficiently and yield both the dispatch decisions and the locational marginal prices (LMPs). The dispatch decisions and LMPs are then tested for their feasibility with respect to the requirements for efficient market-clearing results. The accuracy and scalability of the proposed method is tested on a realistic 1814-bus NYISO system with current and future renewable energy penetration levels.

Exact Fractional Inference via Re-Parametrization & Interpolation between Tree-Re-Weighted- and Belief Propagation- Algorithms

Inference efforts -- required to compute partition function, $Z$, of an Ising model over a graph of $N$ ``spins" -- are most likely exponential in $N$. Efficient variational methods, such as Belief Propagation (BP) and Tree Re-Weighted (TRW) algorithms, compute $Z$ approximately minimizing respective (BP- or TRW-) free energy. We generalize the variational scheme building a $\lambda$-fractional-homotopy, $Z^{(\lambda)}$, where $\lambda=0$ and $\lambda=1$ correspond to TRW- and BP-approximations, respectively, and $Z^{(\lambda)}$ decreases with $\lambda$ monotonically. Moreover, this fractional scheme guarantees that in the attractive (ferromagnetic) case $Z^{(TRW)}\geq Z^{(\lambda)}\geq Z^{(BP)}$, and there exists a unique (``exact") $\lambda_*$ such that, $Z=Z^{(\lambda_*)}$. Generalizing the re-parametrization approach of \cite{wainwright_tree-based_2002} and the loop series approach of \cite{chertkov_loop_2006}, we show how to express $Z$ as a product, $\forall \lambda:\ Z=Z^{(\lambda)}{\cal Z}^{(\lambda)}$, where the multiplicative correction, ${\cal Z}^{(\lambda)}$, is an expectation over a node-independent probability distribution built from node-wise fractional marginals. Our theoretical analysis is complemented by extensive experiments with models from Ising ensembles over planar and random graphs of medium- and large- sizes. The empirical study yields a number of interesting observations, such as (a) ability to estimate ${\cal Z}^{(\lambda)}$ with $O(N^4)$ fractional samples; (b) suppression of $\lambda_*$ fluctuations with increase in $N$ for instances from a particular random Ising ensemble.

Machine Learning for Electricity Market Clearing

This paper seeks to design a machine learning twin of the optimal power flow (OPF) optimization, which is used in market-clearing procedures by wholesale electricity markets. The motivation for the proposed approach stems from the need to obtain the digital twin, which is much faster than the original, while also being sufficiently accurate and producing consistent generation dispatches and locational marginal prices (LMPs), which are primal and dual solutions of the OPF optimization, respectively. Availability of market-clearing tools based on this approach will enable computationally tractable evaluation of multiple dispatch scenarios under a given unit commitment. Rather than direct solution of OPF, the Karush-Kuhn-Tucker (KKT) conditions for the OPF problem in question may be written, and in parallel the LMPs of generators and loads may be expressed in terms of the OPF Lagrangian multipliers. Also, taking advantage of the practical fact that many of the Lagrangian multipliers associated with lines will be zero (thermal limits are not binding), we build and train an ML scheme which maps flexible resources (loads and renewables) to the binding lines, and supplement it with an efficient power-grid aware linear map to optimal dispatch and LMPs. The scheme is validated and illustrated on IEEE models. We also report a trade of analysis between quality of the reconstruction and number of samples needed to train the model.

Model Reduction of Swing Equations with Physics Informed PDE

This manuscript is the first step towards building a robust and efficient model reduction methodology to capture transient dynamics in a transmission level electric power system. Such dynamics is normally modeled on seconds-to-tens-of-seconds time scales by the so-called swing equations, which are ordinary differential equations defined on a spatially discrete model of the power grid. We suggest, following Seymlyen (1974) and Thorpe, Seyler and Phadke (1999), to map the swing equations onto a linear, inhomogeneous Partial Differential Equation (PDE) of parabolic type in two space and one time dimensions with time-independent coefficients and properly defined boundary conditions. The continuous two-dimensional spatial domain is defined by a geographical map of the area served by the power grid, and associated with the PDE coefficients derived from smoothed graph-Laplacian of susceptances, machine inertia and damping. Inhomogeneous source terms represent spatially distributed injection/consumption of power. We illustrate our method on PanTaGruEl (Pan-European Transmission Grid and ELectricity generation model). We show that, when properly coarse-grained, i.e. with the PDE coefficients and source terms extracted from a spatial convolution procedure of the respective discrete coefficients in the swing equations, the resulting PDE reproduces faithfully and efficiently the original swing dynamics. We finally discuss future extensions of this work, where the presented PDE-based reduced modeling will initialize a physics-informed machine learning approach for real-time modeling, $n-1$ feasibility assessment and transient stability analysis of power systems.

Physics Informed Machine Learning of SPH: Machine Learning Lagrangian Turbulence

Smoothed particle hydrodynamics (SPH) is a mesh-free Lagrangian method for obtaining approximate numerical solutions of the equations of fluid dynamics; which has been widely applied to weakly- and strongly compressible turbulence in astrophysics and engineering applications. We present a learn-able hierarchy of parameterized and "physics-explainable" SPH informed fluid simulators using both physics based parameters and Neural Networks (NNs) as universal function approximators. Our learning algorithm develops a mixed mode approach, mixing forward and reverse mode automatic differentiation with forward and adjoint based sensitivity analyses to efficiently perform gradient based optimization. We show that our physics informed learning method is capable of: (a) solving inverse problems over the physically interpretable parameter space, as well as over the space of NN parameters; (b) learning Lagrangian statistics of turbulence (interpolation); (c) combining Lagrangian trajectory based, probabilistic, and Eulerian field based loss functions; and (d) extrapolating beyond training sets into more complex regimes of interest. Furthermore, this hierarchy of models gradually introduces more physical structure, which we show improves interpretability, generalizability (over larger ranges of time scales and Reynolds numbers), preservation of physical symmetries, and requires less training data.

Which Neural Network to Choose for Post-Fault Localization, Dynamic State Estimation and Optimal Measurement Placement in Power Systems?

We consider a power transmission system monitored with Phasor Measurement Units (PMUs) placed at significant, but not all, nodes of the system. Assuming that a sufficient number of distinct single-line faults, specifically pre-fault state and (not cleared) post-fault state, are recorded by the PMUs and are available for training, we, first, design a comprehensive sequence of Neural Networks (NNs) locating the faulty line. Performance of different NNs in the sequence, including Linear Regression, Feed-Forward NN, AlexNet, Graphical Convolutional NN, Neural Linear ODE and Neural Graph-based ODE, ordered according to the type and amount of the power flow physics involved, are compared for different levels of observability. Second, we build a sequence of advanced Power-System-Dynamics-Informed and Neural-ODE based Machine Learning schemes trained, given pre-fault state, to predict the post-fault state and also, in parallel, to estimate system parameters. Finally, third, and continuing to work with the first (fault localization) setting we design a (NN-based) algorithm which discovers optimal PMU placement.

Embedding Power Flow into Machine Learning for Parameter and State Estimation

Modern state and parameter estimations in power systems consist of two stages: the outer problem of minimizing the mismatch between network observation and prediction over the network parameters, and the inner problem of predicting the system state for given values of the parameters. The standard solution of the combined problem is iterative: (a) set the parameters, e.g. to priors on the power line characteristics, (b) map input observation to prediction of the output, (c) compute the mismatch between predicted and observed output, (d) make a gradient descent step in the space of parameters to minimize the mismatch, and loop back to (a). We show how modern Machine Learning (ML), and specifically training guided by automatic differentiation, allows to resolve the iterative loop more efficiently. Moreover, we extend the scheme to the case of incomplete observations, where Phasor Measurement Units (reporting real and reactive powers, voltage and phase) are available only at the generators (PV buses), while loads (PQ buses) report (via SCADA controls) only active and reactive powers. Considering it from the implementation perspective, our methodology of resolving the parameter and state estimation problem can be viewed as embedding of the Power Flow (PF) solver into the training loop of the Machine Learning framework (PyTorch, in this study). We argue that this embedding can help to resolve high-level optimization problems in power system operations and planning.

Message Passing Descent for Efficient Machine Learning

We propose a new iterative optimization method for the {\bf Data-Fitting} (DF) problem in Machine Learning, e.g. Neural Network (NN) training. The approach relies on {\bf Graphical Model} (GM) representation of the DF problem, where variables are fitting parameters and factors are associated with the Input-Output (IO) data. The GM results in the {\bf Belief Propagation} Equations considered in the {\bf Large Deviation Limit} corresponding to the practically important case when the number of the IO samples is much larger than the number of the fitting parameters. We suggest the {\bf Message Passage Descent} algorithm which relies on the piece-wise-polynomial representation of the model DF function. In contrast with the popular gradient descent and related algorithms our MPD algorithm rely on analytic (not automatic) differentiation, while also (and most importantly) it descents through the rugged DF landscape by \emph{making non local updates of the parameters} at each iteration. The non-locality guarantees that the MPD is not trapped in the local-minima, therefore resulting in better performance than locally-updated algorithms of the gradient-descent type. We illustrate superior performance of the algorithm on a Feed-Forward NN with a single hidden layer and a piece-wise-linear activation function.