We develop and test a Lagrangian methodology for extracting finite-cloud geometry and reduced dynamics from particle trajectories. A cloud in a rough, incompressible, two-dimensional synthetic flow is represented by a mass/gyration ellipse describing the particle-weighted bulk and a minimum-area enclosing ellipse describing the outer envelope and its supporting particles. The homogeneous Gaussian–H"older velocity has finite Ornstein–Uhlenbeck memory, and the moving centroid is followed explicitly. The principal finding is unexpected: despite continual deformation by a non-smooth velocity field, the normalized aspect ratios of both ellipses reach broad, order-one distributions that are approximately stationary when conditioned on cloud scale. The mass ellipse is more elongated, whereas the enclosing ellipse spans a larger radial envelope and has the more scale-stable conditional shape distribution. To understand this saturation, we compare two finite-cloud descriptions of the velocity gradient. A gradient averaged over the enclosing ellipse predicts more aligned stretching than the particles experience. The resulting gradient correction supplies most of the negative contribution offsetting aligned stretching, while a smaller correction is specific to representing the evolving outer cloud by an enclosing ellipse. Finally, we demonstrate a data-to-model workflow. The measured time series is transformed to intrinsic variables, a constrained hierarchy of finite-lag stochastic models is fitted, and the models are compared on held-out realizations. The fitted coefficients and equations are specific to this synthetic experiment. The transferable contribution is the methodology: paired cloud geometries, finite-cloud coarse graining, intrinsic variables, transparent correction decomposition, and held-out validation for future simulations and experiments.

Most modern bridge-diffusion methods achieve finite-time transport by specifying an interpolation, Schrödinger-bridge, or stochastic-control objective and then learning the associated score or drift field with a neural network. In contrast, we identify a restricted but sufficiently broad and analytically solvable class in which the score, intermediate marginals, and protocol gradients are available in closed form without inner stochastic simulation loops and without neural networks in the optimization loop. We recast the classical linear–quadratic–Gaussian (LQG) stochastic-control structure as a transport problem of the Path Integral Diffusion (PID) type. In classical LQG control, linear dynamics, Gaussian noise, and quadratic costs lead to Riccati equations and closed-form optimal feedback. In LQ-GM-PID, we retain the linear–quadratic stochastic-control backbone, but replace terminal state regulation by a prescribed terminal probability density and allow both the initial and terminal laws to be Gaussian Mixtures (GM). Moreover, LQ-GM-PID turns bridge diffusion from a tool for terminal target matching alone into a tool for path shaping. We demonstrate this on a 2D corridor task, a 2D multi-entrance transport task, and a high-dimensional scaling study with d=32 and M=16 Gaussian-mixture terminal modes, all with sub-50 ms analytic precompute on a laptop. We position LQ-GM-PID as an analytically solvable reference model for the state-of-the-art neural bridge-diffusion and generative-transport methods: a controlled setting in which neural approximations, score estimates, path-shaping objectives, and protocol-learning procedures can be tested against exact quantities.

Harmonic Path Integral Diffusion (H-PID) provides an analytically tractable framework for sampling from a target density p(tar)(x)∝exp(-E(x)). H-PID can be viewed as a diffusion bridge model solving a stochastic optimal transport problem from a δ-density at t=0 to the target density at t=1. The dynamics are governed by a controlled stochastic differential equation, and the corresponding variational stochastic optimal transport objective combines a time-dependent quadratic potential, βt∥xt∥2/2, with a kinetic control cost, ∥u(t;xt)∥2/2. The focus of this paper is the design of the temporal stiffness protocol βt, which enables explicit control of intermediate sampling dynamics when the terminal density is fixed. We exploit the central advantage of H-PID-its integrability-which yields an explicit representation of the optimal control in terms of the target density and Green functions of the associated linear forward and backward diffusion-in-a-potential problems. Our main contribution is to convert this integrable structure into a practical methodology for protocol optimization. Specializing to piecewise-constant stiffness schedules and Gaussian-mixture targets, we develop two complementary optimization principles: The first is a deterministic one, relying on explicit evaluation of the dynamic marginals, and exemplified on a velocity-gradient-sensitivity objective, which provides a computationally controlled framework for optimizing transport regularity and stiffness. The second is a stochastic one, implemented via sampling, and exemplified on sharpness-based temporal-memory objective regularized to favor transitions within a prescribed time window that targets the temporal organization of the sampling path. These two objectives illuminate different aspects of the same protocol-design problem. The velocity-gradient-sensitivity objective serves as a clean methodological backbone and supports interpretable optimization and scaling studies. The sharpness-based objective reveals that schedule quality is target-dependent, and that the dependence on β is not universal: different target geometries may favor different stiffness regimes and qualitatively different transient organizations. Examples with low- and moderate-dimensional Gaussian mixtures demonstrate that the proposed approach can control not only the terminal sampling accuracy but also the transient evolution of probability mass, while remaining computationally light and theoretically transparent.

Independent sample generation is the prevailing paradigm in modern diffusion-based generative models of AI. We ask a different question: can samples coordinate through shared population statistics to transport probability mass more efficiently? We introduce Mean-Field Path-Integral Diffusion (MF-PID), a framework in which samples are promoted to interacting agents whose drift depends self-consistently on the evolving population density. The coupling converts distribution matching into a McKean–Vlasov extension of the stochastic optimal transport problem, unifying generative modeling and multi-agent control under the same Hamilton–Jacobi–Bellman/Kolmogorov–Fokker–Planck duality. We identify two analytically tractable regimes: a Linear–Quadratic–Gaussian (LQG) benchmark in which the infinite-dimensional mean-field system reduces to a finite set of Riccati and linear ODEs, and a Gaussian-mixture regime governed by a piecewise-constant protocol that preserves closed-form solvability. For a quadratic interaction potential with schedule β_t and zero base drift we prove that the self-consistent MF guidance is the exact linear interpolant between initial and target global means – a result that holds for arbitrary initial and target densities and any β_t. Applied to demand-response control of energy systems, where agents aggregated into an ensemble are energy consumers (e.g. thermal zones within a building), MF-PID achieves 19–24% reductions in cumulative control energy over independent-agent baselines while matching the prescribed terminal distribution exactly, and reveals how coordination redistributes actuation effort across heterogeneous sub-populations.

An agent that operates sequentially must incorporate new experience without forgetting old experience, under a fixed memory budget. We propose a framework in which memory is not a parameter vector but a stochastic process: a Bridge Diffusion on a replay interval [0,1], whose terminal marginal encodes the present and whose intermediate marginals encode the past. New experience is incorporated via a three-step Compress–Add–Smooth (CAS) recursion. We test the framework on the class of models with marginal probability densities modeled via Gaussian mixtures of fixed number of components K in d dimensions; temporal complexity is controlled by a fixed number L of piecewise-linear protocol segments whose nodes store Gaussian-mixture states. The entire recursion costs O(LKd^2) flops per day – no backpropagation, no stored data, no neural networks – making it viable for controller-light hardware. Forgetting in this framework arises not from parameter interference but from lossy temporal compression: the re-approximation of a finer protocol by a coarser one under a fixed segment budget. We find that the retention half-life scales linearly as a_1/2≈ c L with a constant c>1 that depends on the dynamics but not on the mixture complexity K, the dimension d, or the geometry of the target family. The constant c admits an information-theoretic interpretation analogous to the Shannon channel capacity. The stochastic process underlying the bridge provides temporally coherent “movie” replay – compressed narratives of the agent's history, demonstrated visually on an MNIST latent-space illustration. The framework provides a fully analytical “Ising model” of continual learning in which the mechanism, rate, and form of forgetting can be studied with mathematical precision.

Direct numerical simulation of turbulence at realistic Reynolds numbers is still beyond current computational capability, necessitating models that reduce the number of resolved spatial scales. Motivated by phenomenology and recent data-driven works based on universality of the smallest scales in fully developed turbulence, the statistical dynamics of the velocity gradient tensor (VGT) at the Kolmogorov scale become of critical importance in advancing turbulence models. Physics-informed machine learning has found considerable success in exploiting large datasets taken from direct numerical simulation of Navier-Stokes to improve models for the evolution of the VGT. In this work, we follow the long line of blending physical insight with data analysis to simultaneously advance both the modeling and understanding of the phenomenology of the VGT. Using the intimate connection between VGT evolution and fluid deformation, we develop the Lagrangian attention tensor network approach that significantly improves over current physics-informed machine learning methods. We demonstrate state-of-the-art performance in both a-priori and a-posteriori metrics, before interpreting the trained attention mechanisms to discover a surprising connection between the history of the strain-rate-tensor and the pressure Hessian.

Harmonic Path Integral Diffusion (H-PID) introduces a novel approach to sampling from complex, continuous probability distributions by creating a time-dependent “bridge” from an initial point to the target distribution. Formulated as a Stochastic Optimal Control problem, H-PID balances control effort and accuracy through a unique three-level integrable structure: Top Level: Potential, force, and gauge terms combine to form a linearly solvable Path Integral Control system based on Green functions. Mid Level:With quadratic potentials and affine force/gauge terms, the Green functions reduce to Gaussian forms, mirroring quantum harmonic oscillators in imaginary time. Bottom Level: For a uniform quadratic case, the optimal drift/control reduces to a convolution of the target distribution with a Gaussian kernel, enabling efficient sampling. Implementation-wise the low-level H-PID operates without neural networks, allowing it to run efficiently on standard CPUs while achieving high precision. Validated on Gaussian mixtures and CIFAR-10 images, H-PID reveals a “weighted state” parameter as an order parameter in a dynamic phase transition, signaling early completion of the sampling process. This feature positions H-PID as a strong alternative to traditional methods sampling, such as simulated annealing, particularly for applications that demand analytical control, computational efficiency, and scalability.

Diffusion-based samplers – Score Based Diffusions, Bridge Diffusions and Path Integral Diffusions – match a target at terminal time, but the real leverage comes from choosing the schedule that governs the intermediate-time dynamics. We develop a path-wise schedule – selection gramework for Harmonic PID with a time-varying stiffness, exploiting Piece-Wise-Constant(PWC) parametrizations and a simple hierarchical refinement. We introduce schedule-sensitive Quality-of-Sampling (QoS) diagnostics. Assuming a Gaussian-Mixture (GM) target, we retain closed-form Green functions' ration and numerically stable, Neural-Network free oracles for predicted-state maps and score. Experiments in 2D show that QoS driven PWC schedules consistently improve early-exit fidelity, tail accuracy, conditioning of the dynamics, and speciation (label-selection) timing at fixed integration budgets.

We introduce Guided Harmonic Path-Integral Diffusion (GH-PID), a linearly-solvable framework for guided Stochastic Optimal Transport (SOT) with a hard terminal distribution and soft, application-driven path costs. A low-dimensional guidance protocol shapes the trajectory ensemble while preserving analytic structure: the forward and backward Kolmogorov equations remain linear, the optimal score admits an explicit Green-function ratio, and Gaussian-Mixture Model (GMM) terminal laws yield closed-form expressions. This enables stable sampling and differentiable protocol learning under exact terminal matching. We develop guidance-centric diagnostics – path cost, centerline adherence, variance flow, and drift effort – that make GH-PID an interpretable variational ansatz for empirical SOT. Three navigation scenarios illustrated in 2D: (i) Case A: hand-crafted protocols revealing how geometry and stiffness shape lag, curvature effects, and mode evolution; (ii) Case B: single-task protocol learning, where a PWC centerline is optimized to minimize integrated cost; (iii) Case C: multi-expert fusion, in which a commander reconciles competing expert/teacher trajectories and terminal beliefs through an exact product-of-experts law and learns a consensus protocol. Across all settings, GH-PID generates geometry-aware, trust-aware trajectories that satisfy the prescribed terminal distribution while systematically reducing integrated cost.

Scientific generative models must turn tractable local decisions into globally correlated samples that respect physical constraints. We introduce Sampling Decisions, a finite-horizon framework in which a structured object is assembled on a growing state graph and corrected globally by an exact path-space control law. For a prescribed Gibbs target, the corrected law is the unique relative-entropy projection of a sequential prior and is realized by a Doob h-transform with a linear backward desirability recursion. The same object admits equivalent interpretations as a KL-optimal controller, a one-sided Schrodinger transport, and an ideal value or flow function for autoregressive and GFlowNet-type generation. A route-resolved formulation yields a finite-particle algorithm, whose transition kernels and terminal law converge as the path budget grows. For binary graphical models, we prove a structural cancellation theorem: every fixed singleton-product prior disappears from the population correction, so improved one-point marginals do not alter the exact generative dynamics. The relevant information is conditional and prefix dependent. Statistical physics supplies this structure through a Local-Boltzmann prior that absorbs interactions as spins are revealed, while optimal path-space control supplies the missing look-ahead field from the unrevealed subgraph. On exactly enumerable Ising grids, this physics-informed proposal increases effective sample size by factors of about ten to nearly one thousand relative to product proposals and reaches the exact-target reference band at the tested budgets. On a larger 10*10 grid, beyond exact enumeration, the same hierarchy persists: Local-Boltzmann guidance avoids the severe weight collapse of product proposals and approaches a long-run MCMC baseline on the reported diagnostics.

Turbulent diffusion causes particles placed in proximity to separate. We investigate the required swimming efforts to maintain an active particle close to its passively advected counterpart. We explore optimally balancing these efforts by developing a novel physics-informed reinforcement learning strategy and comparing it with prescribed control and physics-agnostic reinforcement learning strategies. Our scheme, coined the actor-physicist, is an adaptation of the actor-critic algorithm in which the neural network parameterized critic is replaced with an analytically derived physical heuristic function, the physicist. We validate the proposed physics-informed reinforcement learning approach through extensive numerical experiments in both synthetic Batchelor-Kraichnan and more realistic Arnold-Beltrami-Childress flow environments, demonstrating its superiority in controlling particle dynamics when compared to standard reinforcement learning methods.

Abstract. We study the Floating Offshore Wind Turbine (FOWT) dynamic response to high wind velocity scenarios utilizing an extensive Markov Chain Monte Carlo simulation involving 10,000 trials of a reduced model from Betti et al. (2014) with a blade-pitch PID controller. The research emphasizes analysis of extreme events in surge, pitch and heave of FOWT, identifying and categorizing them based on their statistics, correlations and causal relations to wind and waves. A significant insight is the differentiation of anomalies into short-correlated and long-correlated types, with the latter primarily influenced by wind conditions. For example, one of our findings is that while specific wave conditions may amplify anomalies, wind remains the predominant factor in long-term anomalous pitch behaviors. We anticipate that further development of this analysis of the operation's critical extreme events will be pivotal for advancing control strategies and design considerations in FOWTs, accommodating the turbulent environment they encounter.

In many power systems, particularly those isolated from larger intercontinental grids, operational dependence on natural gas becomes pivotal, especially during fluctuations or unavailability of renewables coupled with uncertain consumption patterns. Efficient orchestration and inventive strategies are imperative for the smooth functioning of these standalone gas-grid systems. This paper delves into the challenge of synchronized dynamic and stochastic optimization for independent transmission-level gas-grid systems. Our approach's novelty lies in amalgamating the staggered-grid method for the direct assimilation of gas-flow PDEs with an automated sensitivity analysis facilitated by SciML/Julia, further enhanced by an intuitive linkage between gas and power grids via nodal flows. We initiate with a single pipe to establish a versatile and expandable methodology, later showcasing its effectiveness with increasingly intricate examples.

Energy efficiency of buildings is considered to be one of the major means of achieving the net-zero carbon goal around the world. The big part of the energy savings are supposed to be coming from optimizing the operation of the building heating, ventilation, and air conditioning (HVAC) systems. There is a natural trade-off between the energy efficiency and the indoor comfort level, and finding an optimal operating schedule/regime requires knowing the occupancy of different spaces inside of the building. Moreover, the COVID-19 pandemic has also revealed the need to sustain the high quality of the indoor air in order to reduce the risk of spread of infection. Occupancy detection from indoor sensors is thus an important practical problem. In the present paper, we propose detection of occupancy based on the carbon dioxide measurements inside the building. In particular, a new approach based on the, so-called, switching auto-regressive process with Markov regime is presented and justified by the physical model of the carbon dioxide concentration dynamics. We demonstrate the efficiency of the method compared to simple Hidden Markov approaches on simulated and real-life data. We also show that the model is flexible and can be generalized to account for different ventilation regimes, simultaneously detecting the occupancy and the ventilation rate.

Faced with the complexities of managing natural gas-dependent power system amid the surge of renewable integration and load unpredictability, this study explores strategies for navigating emergency transitions to costlier secondary fuels. Our aim is to develop decision-support tools for operators during such exigencies. We approach the problem through a Markov Decision Process (MDP) framework, accounting for multiple uncertainties. These include the potential for dual-fuel generator failures and operator response during high-pressure situations. Additionally, we consider the finite reserves of primary fuel, governed by gas-flow partial differential equations (PDEs) and constrained by nodal pressure. Other factors include the variability in power forecasts due to renewable generation and the economic impact of compulsory load shedding. For tractability, we address the MDP in a simplified context, replacing it by Markov Processes evaluated against a selection of policies and scenarios for comparison. Our study considers two models for the natural gas system: an over-simplified model tracking linepack and a more nuanced model that accounts for gas flow network heterogeneity. The efficacy of our methods is demonstrated using a realistic model replicating Israel's power-gas infrastructure.

In this study, we introduce a novel method for generating i.i.d. synthetic samples from high-dimensional real-valued probability distributions, defined by Ground Truth (GT) samples. Our approach integrates space-time mixing strategies across temporal and spatial dimensions. The methodology involves three interrelated stochastic processes: (a) linear processes with space-time mixing yielding Gaussian conditional densities, (b) their diffusion bridge analogs conditioned on initial and final states, and (c) nonlinear stochastic processes refined via score-matching techniques. The training regime fine-tunes both nonlinear and potentially linear models to align with GT data. We validate our space-time diffusion bridge approach through numerical experiments. Copyright (c) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)

The computational complexity of inference -- required to compute the partition function, $Z$, of an Ising model over a graph of $N$''spins" -- is most likely exponential in $N$. Efficient variational methods, such as Belief Propagation (BP) and Tree Re-Weighted (TRW) algorithms, compute $Z$ approximately by minimizing the respective (BP- or TRW-) free energy. We generalize the variational scheme by building a $\lambda$-fractional interpolation, $Z^{(\lambda)}$, where $\lambda=0$ and $\lambda=1$ correspond to TRW- and BP-approximations, respectively. This fractional scheme -- coined Fractional Belief Propagation (FBP) -- 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 \citep{wainwright_tree-based_2002} and the loop series approach of \citep{chertkov_loop_2006}, we show how to express $Z$ as a product, $\forall \lambda:\ Z=Z^{(\lambda)}{\tilde Z}^{(\lambda)}$, where the multiplicative correction, ${\tilde Z}^{(\lambda)}$, 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 the ability to estimate ${\tilde Z}^{(\lambda)}$ with $O(N^{2::4})$ fractional samples and suppression of $\lambda_*$ fluctuations with an increase in $N$ for instances from a particular random Ising ensemble. We also verify and discuss the applicability of this approach to the problem of image de-noising.