Lagrangian large eddy simulations via physics-informed machine learning.

High-Reynolds number homogeneous isotropic turbulence (HIT) is fully described within the Navier-Stokes (NS) equations, which are notoriously difficult to solve numerically. Engineers, interested primarily in describing turbulence at a reduced range of resolved scales, have designed heuristics, known as large eddy simulation (LES). LES is described in terms of the temporally evolving Eulerian velocity field defined over a spatial grid with the mean-spacing correspondent to the resolved scale. This classic Eulerian LES depends on assumptions about effects of subgrid scales on the resolved scales. Here, we take an alternative approach and design LES heuristics stated in terms of Lagrangian particles moving with the flow. Our Lagrangian LES, thus L-LES, is described by equations generalizing the weakly compressible smoothed particle hydrodynamics formulation with extended parametric and functional freedom, which is then resolved via Machine Learning training on Lagrangian data from direct numerical simulations of the NS equations. The L-LES model includes physics-informed parameterization and functional form, by combining physics-based parameters and physics-inspired Neural Networks to describe the evolution of turbulence within the resolved range of scales. The subgrid-scale contributions are modeled separately with physical constraints to account for the effects from unresolved scales. We build the resulting model under the differentiable programming framework to facilitate efficient training. We experiment with loss functions of different types, including physics-informed ones accounting for statistics of Lagrangian particles. We show that our L-LES model is capable of reproducing Eulerian and unique Lagrangian turbulence structures and statistics over a range of turbulent Mach numbers.

Statistical mechanics of thermostatically controlled multizone buildings.

We study the collective phenomena and constraints associated with the aggregation of individual cooling units from a statistical mechanics perspective. These units are modeled as thermostatically controlled loads (TCLs) and represent zones in a large commercial or residential building. Their energy input is centralized and controlled by a collective unit-the air handling unit (AHU)-delivering cool air to all TCLs, thereby coupling them together. Aiming to identify representative qualitative features of the AHU-to-TCL coupling, we build a simple but realistic model and analyze it in two distinct regimes: the constant supply temperature (CST) and the constant power input (CPI) regimes. In both cases, we center our analysis on the relaxation dynamics of individual TCL temperatures to a statistical steady state. We observe that while the dynamics are relatively fast in the CST regime, resulting in all TCLs evolving around the control set point, the CPI regime reveals the emergence of a bimodal probability distribution and two, possibly strongly separated, timescales. We observe that the two modes in the CPI regime are associated with all TCLs being in the same low or high airflow states, with an occasional collective transition between the modes akin to Kramer's phenomenon in statistical physics. To the best of our knowledge, this phenomenon has been overlooked in building energy systems despite its direct operational implications. It highlights a trade-off between occupational comfort-related to zonal temperature variations-and energy consumption.

Prediction and prevention of pandemics via graphical model inference and convex programming.

Hard-to-predict bursts of COVID-19 pandemic revealed significance of statistical modeling which would resolve spatio-temporal correlations over geographical areas, for example spread of the infection over a city with census tract granularity. In this manuscript, we provide algorithmic answers to the following two inter-related public health challenges of immense social impact which have not been adequately addressed (1) Inference Challenge assuming that there are N census blocks (nodes) in the city, and given an initial infection at any set of nodes, e.g. any N of possible single node infections, any [Formula: see text] of possible two node infections, etc, what is the probability for a subset of census blocks to become infected by the time the spread of the infection burst is stabilized? (2) Prevention Challenge What is the minimal control action one can take to minimize the infected part of the stabilized state footprint? To answer the challenges, we build a Graphical Model of pandemic of the attractive Ising (pair-wise, binary) type, where each node represents a census tract and each edge factor represents the strength of the pairwise interaction between a pair of nodes, e.g. representing the inter-node travel, road closure and related, and each local bias/field represents the community level of immunization, acceptance of the social distance and mask wearing practice, etc. Resolving the Inference Challenge requires finding the Maximum-A-Posteriory (MAP), i.e. most probable, state of the Ising Model constrained to the set of initially infected nodes. (An infected node is in the [Formula: see text] state and a node which remained safe is in the [Formula: see text] state.) We show that almost all attractive Ising Models on dense graphs result in either of the two possibilities (modes) for the MAP state: either all nodes which were not infected initially became infected, or all the initially uninfected nodes remain uninfected (susceptible). This bi-modal solution of the Inference Challenge allows us to re-state the Prevention Challenge as the following tractable convex programming: for the bare Ising Model with pair-wise and bias factors representing the system without prevention measures, such that the MAP state is fully infected for at least one of the initial infection patterns, find the closest, for example in [Formula: see text], [Formula: see text] or any other convexity-preserving norm, therefore prevention-optimal, set of factors resulting in all the MAP states of the Ising model, with the optimal prevention measures applied, to become safe. We have illustrated efficiency of the scheme on a quasi-realistic model of Seattle. Our experiments have also revealed useful features, such as sparsity of the prevention solution in the case of the [Formula: see text] norm, and also somehow unexpected features, such as localization of the sparse prevention solution at pair-wise links which are NOT these which are most utilized/traveled.

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 using 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 the pre-fault state and the (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 NNs, AlexNet, graph convolutional NNs, neural linear ordinary differential equations (ODEs) and neural graph-based ODEs, 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 that are trained, given the 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 an (NN-based) algorithm which discovers optimal PMU placement.

Mean-field control for efficient mixing of energy loads.

We pose an engineering challenge of controlling an ensemble of energy devices via coordinated, implementation-light, and randomized on/off switching as a problem in nonequilibrium statistical mechanics. We show that mean-field control with nonlinear feedback on the cumulative consumption, assumed available to the aggregator via direct physical measurements of the energy flow, allows the ensemble to recover from its use in the demand response regime, i.e., transition to a statistical steady state, significantly faster than in the case of the fixed feedback. Moreover when the nonlinearity is sufficiently strong, one observes the phenomenon of "super-relaxation," where the total instantaneous energy consumption of the ensemble transitions to the steady state much faster than the underlying probability distribution of the devices over their state space, while also leaving almost no devices outside of the comfort zone.

Power of Ensemble Diversity and Randomization for Energy Aggregation.

We study an ensemble of diverse (inhomogeneous) thermostatically controlled loads aggregated to provide the demand response (DR) services in a district-level energy system. Each load in the ensemble is assumed to be equipped with a random number generator switching heating/cooling on or off with a Poisson rate, r, when the load leaves the comfort zone. Ensemble diversity is modeled through inhomogeneity/disorder in the deterministic dynamics of loads. Approached from the standpoint of statistical physics, the ensemble represents a non-equilibrium system driven away from its natural steady state by the DR. The ability of the ensemble to recover by mixing faster to the steady state after its DR's use is advantageous. The trade-off between the level of the aggregator's control, commanding the devices to lower the rate r, and the phase-space-oscillatory deterministic dynamics is analyzed. Then, we study the effect of the load diversity, investigating four different disorder probability distributions (DPDs) ranging from the case of the Gaussian DPD to the case of the uniform with finite support DPD. We show that stronger regularity of the DPD results in faster mixing, which is similar to the Landau damping in plasma physics. Our theoretical analysis is supported by extensive numerical validation.

Optimal structure and parameter learning of Ising models.

Reconstruction of the structure and parameters of an Ising model from binary samples is a problem of practical importance in a variety of disciplines, ranging from statistical physics and computational biology to image processing and machine learning. The focus of the research community shifted toward developing universal reconstruction algorithms that are both computationally efficient and require the minimal amount of expensive data. We introduce a new method, interaction screening, which accurately estimates model parameters using local optimization problems. The algorithm provably achieves perfect graph structure recovery with an information-theoretically optimal number of samples, notably in the low-temperature regime, which is known to be the hardest for learning. The efficacy of interaction screening is assessed through extensive numerical tests on synthetic Ising models of various topologies with different types of interactions, as well as on real data produced by a D-Wave quantum computer. This study shows that the interaction screening method is an exact, tractable, and optimal technique that universally solves the inverse Ising problem.

Ensemble of Thermostatically Controlled Loads: Statistical Physics Approach.

Thermostatically controlled loads, e.g., air conditioners and heaters, are by far the most widespread consumers of electricity. Normally the devices are calibrated to provide the so-called bang-bang control - changing from on to off, and vice versa, depending on temperature. We considered aggregation of a large group of similar devices into a statistical ensemble, where the devices operate following the same dynamics, subject to stochastic perturbations and randomized, Poisson on/off switching policy. Using theoretical and computational tools of statistical physics, we analyzed how the ensemble relaxes to a stationary distribution and established a relationship between the relaxation and the statistics of the probability flux associated with devices' cycling in the mixed (discrete, switch on/off, and continuous temperature) phase space. This allowed us to derive the spectrum of the non-equilibrium (detailed balance broken) statistical system and uncover how switching policy affects oscillatory trends and the speed of the relaxation. Relaxation of the ensemble is of practical interest because it describes how the ensemble recovers from significant perturbations, e.g., forced temporary switching off aimed at utilizing the flexibility of the ensemble to provide "demand response" services to change consumption temporarily to balance a larger power grid. We discuss how the statistical analysis can guide further development of the emerging demand response technology.

Fault-induced delayed voltage recovery in a long inhomogeneous power-distribution feeder.

We analyze the dynamics of a distribution circuit loaded with many induction motors and subjected to sudden changes in voltage at the beginning of the circuit. As opposed to earlier work by Duclut et al. [Phys. Rev. E 87, 062802 (2013)], the motors are disordered, i.e., the mechanical torque applied to the motors varies in a random manner along the circuit. In spite of the disorder, many of the qualitative features of a homogeneous circuit persist, e.g., long-range motor-motor interactions mediated by circuit voltage and electrical power flows result in coexistence of the spatially extended and propagating normal and stalled phases. We also observed a new phenomenon absent in the case without inhomogeneity or disorder. Specifically, the transition front between the normal and stalled phases becomes somewhat random, even when the front is moving very slowly or is even stationary. Motors within the blurred domain appear in a normal or stalled state depending on the local configuration of the disorder. We quantify the effects of the disorder and discuss the statistics of distribution dynamics, e.g., the front position and width, total active and reactive consumption of the feeder, and maximum clearing time.

Hysteresis, phase transitions, and dangerous transients in electrical power distribution systems.

The majority of dynamical studies in power systems focus on the high-voltage transmission grids where models consider large generators interacting with crude aggregations of individual small loads. However, new phenomena have been observed indicating that the spatial distribution of collective, nonlinear contribution of these small loads in the low-voltage distribution grid is crucial to the outcome of these dynamical transients. To elucidate the phenomenon, we study the dynamics of voltage and power flows in a spatially extended distribution feeder (circuit) connecting many asynchronous induction motors and discover that this relatively simple 1+1 (space+time) dimensional system exhibits a plethora of nontrivial spatiotemporal effects, some of which may be dangerous for power system stability. Long-range motor-motor interactions mediated by circuit voltage and electrical power flows result in coexistence and segregation of spatially extended phases defined by individual motor states, a "normal" state where the motors' mechanical (rotation) frequency is slightly smaller than the nominal frequency of the basic ac flows and a "stalled" state where the mechanical frequency is small. Transitions between the two states can be initiated by a perturbation of the voltage or base frequency at the head of the distribution feeder. Such behavior is typical of first-order phase transitions in physics, and this 1+1 dimensional model shows many other properties of a first-order phase transition with the spatial distribution of the motors' mechanical frequency playing the role of the order parameter. In particular, we observe (a) propagation of the phase-transition front with the constant speed (in very long feeders) and (b) hysteresis in transitions between the normal and stalled (or partially stalled) phases.

Synchronization in complex oscillator networks and smart grids.

The emergence of synchronization in a network of coupled oscillators is a fascinating topic in various scientific disciplines. A widely adopted model of a coupled oscillator network is characterized by a population of heterogeneous phase oscillators, a graph describing the interaction among them, and diffusive and sinusoidal coupling. It is known that a strongly coupled and sufficiently homogeneous network synchronizes, but the exact threshold from incoherence to synchrony is unknown. Here, we present a unique, concise, and closed-form condition for synchronization of the fully nonlinear, nonequilibrium, and dynamic network. Our synchronization condition can be stated elegantly in terms of the network topology and parameters or equivalently in terms of an intuitive, linear, and static auxiliary system. Our results significantly improve upon the existing conditions advocated thus far, they are provably exact for various interesting network topologies and parameters; they are statistically correct for almost all networks; and they can be applied equally to synchronization phenomena arising in physics and biology as well as in engineered oscillator networks, such as electrical power networks. We illustrate the validity, the accuracy, and the practical applicability of our results in complex network scenarios and in smart grid applications.

Message passing for optimization and control of a power grid: model of a distribution system with redundancy.

We use a power grid model with M generators and N consumption units to optimize the grid and its control. Each consumer demand is drawn from a predefined finite-size-support distribution, thus simulating the instantaneous load fluctuations. Each generator has a maximum power capability. A generator is not overloaded if the sum of the loads of consumers connected to a generator does not exceed its maximum production. In the standard grid each consumer is connected only to its designated generator, while we consider a more general organization of the grid allowing each consumer to select one generator depending on the load from a predefined consumer dependent and sufficiently small set of generators which can all serve the load. The model grid is interconnected in a graph with loops, drawn from an ensemble of random bipartite graphs, while each allowed configuration of loaded links represent a set of graph covering trees. Losses, the reactive character of the grid and the transmission-level connections between generators (and many other details relevant to realistic power grid) are ignored in this proof-of-principles study. We focus on the asymptotic limit, N-->infinity and N/M-->D=O(1)>1 , and we show that the interconnects allow significant expansion of the parameter domains for which the probability of a generator overload is asymptotically zero. Our construction explores the formal relation between the problem of grid optimization and the modern theory of sparse graphical models. We also design heuristic algorithms that achieve the asymptotically optimal selection of loaded links. We conclude discussing the ability of this approach to include other effects such as a more realistic modeling of the power grid and related optimization and control algorithms.

Loop calculus in statistical physics and information science.

Considering a discrete and finite statistical model of a general position we introduce an exact expression for the partition function in terms of a finite series. The leading term in the series is the Bethe-Peierls (belief propagation) (BP) contribution; the rest are expressed as loop contributions on the factor graph and calculated directly using the BP solution. The series unveils a small parameter that often makes the BP approximation so successful. Applications of the loop calculus in statistical physics and information science are discussed.

Extreme outages caused by polarization mode dispersion.

We study the dependence on fiber birefringence of the bit-error rate (BER) caused by amplifier noise in a linear optical fiber telecommunication system. We show that the probability-distribution function of the BER obtained by averaging over many realizations of birefringent disorder has an extended tail that corresponds to anomalously large values of BER. We specifically discuss the dependence of the tail on such details of pulse detection at the fiber output as setting the clock and filtering procedures.

Phenomenology of Rayleigh-Taylor turbulence.

I analyze the advanced mixing regime of the Rayleigh-Taylor incompressible turbulence in the small Atwood number Boussinesq approximation. The prime focus of my phenomenological approach is to resolve the temporal behavior and the small-scale spatial correlations of velocity and temperature fields inside the mixing zone, which grows as proportional, variant t(2). I show that the "5/3"-Kolmogorov scenario for velocity and temperature spectra is realized in three spatial dimensions with the viscous and dissipative scales decreasing in time, proportional, variant t(-1/4). The Bolgiano-Obukhov scenario is shown to be valid in two dimensions with the viscous and dissipative scales growing, proportional, variant t(1/8).

Compensation for extreme outages caused by polarization mode dispersion and amplifier noise.

Fluctuations of Bit-Error-Rate (BER) stimulated by birefringent disorder in an optical fiber system are found to be strong. The effect may not be analyzed in terms of the average BER but rather requires analyzing the Probability Distribution Function (PDF) of BER. We report the emergence of the extremely extended algebraic-like tail of the PDF, corresponding to anomalously large values of BER. We analyze the dependence of the PDF tail, and thus the outage probability, on the first-order PMD compensation scheme. Effectiveness of compensation is illustrated quantitatively using a simple, however, practical example.