The Fokker-Planck (FP) equation is a foundational PDE in stochastic processes. However, curse of dimensionality (CoD) poses challenge when dealing with high-dimensional FP PDEs. Although Monte Carlo and vanilla Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimension increases, surpassing the precision of numerical simulations and resulting in substantial errors. Moreover, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a difficult HJB equation, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in SDEs. The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling. Three fitting methods, Score Matching (SM), Sliced SM (SSM), and Score-PINN, are introduced. The proposed score-based SDE solver operates in two stages: first, employing SM, SSM, or Score-PINN to acquire the score; and second, solving the LL via an ODE using the obtained score. Comparative evaluations across these methods showcase varying trade-offs. The proposed method is evaluated across diverse SDEs, including anisotropic OU processes, geometric Brownian, and Brownian with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy. The numerical results demonstrate the score-based SDE solver's stability, speed, and performance across different settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

Machine learning interatomic potentials are revolutionizing large-scale, accurate atomistic modelling in material science and chemistry. These potentials often use atomic cluster expansion or equivariant message passing with spherical harmonics as basis functions. However, the dependence on Clebsch-Gordan coefficients for maintaining rotational symmetry leads to computational inefficiencies and redundancies. We propose an alternative: a Cartesian-coordinates-based atomic density expansion. This approach provides a complete description of atomic environments while maintaining interaction body orders. Additionally, we integrate low-dimensional embeddings of various chemical elements and inter-atomic message passing. The resulting potential, named Cartesian Atomic Cluster Expansion (CACE), exhibits good accuracy, stability, and generalizability. We validate its performance in diverse systems, including bulk water, small molecules, and 25-element high-entropy alloys.

Subsea exploration, inspection, and intervention operations heavily rely on remotely operated vehicles (ROVs). However, the inherent complexity of the underwater environment presents significant challenges to the operators of these vehicles. This paper delves into the challenges associated with navigation and maneuvering tasks in the teleoperation of ROVs, such as reduced situational awareness and heightened teleoperator workload. To address these challenges, we introduce an underwater Digital Twin (DT) system designed to enhance underwater teleoperation, enable autonomous navigation, support system monitoring, and facilitate system testing through simulation. Our approach involves a dynamic representation of the underwater robot and its environment using desktop virtual reality, as well as the integration of mapping, localization, path planning and simulation capabilities within the DT system. Our research demonstrates the system's adaptability, versatility and feasibility, highlighting significant challenges and, in turn, improving the teleoperators' situational awareness and reducing their workload.

In supervised learning of neural networks and regression models, understanding the dynamics of optimization algorithms, and in particular stochastic gradient descent (SGD), is of utmost importance. However, despite much progress in a number of directions, this still remains a highly challenging theoretical problem. A fruitful approach that allows making analytical progress consists of suitably approximating SGD by a continuous time approximation, henceforth called stochastic gradient flow (SGF). In this contribution, we build up on this approach, to develop a general formalism characterizing the dynamics of the stochastic process, and apply it to the investigation of the test risk (or generalization error) as a function of time. As is well known, the classical bias-variance trade-off has been challenged in a number of models displaying the double descent phenomenon [1, 2, 3]. Analytical derivations of double descent curves have been achieved for relatively simple models, but are limited to the use of least squares estimators (no dynamics) and pure gradient flow (GF) approximations of gradient descent (GD). The present work goes one step further by investigating the effects of stochasticity on the double descent curve. Our main contributions are summarized as follows: C1 We consider a general Itô stochastic differential equation (SDE) and represent the Markovian transition probability as a path integral, Eq. (12). A general'explicit' formula for the transition probability, Eq. (18), is derived in the limit of a small learning rate by using a Laplace approximation.

Automated and autonomous industrial inspection is a longstanding research field, driven by the necessity to enhance safety and efficiency within industrial settings. In addressing this need, we introduce an autonomously navigating robotic system designed for comprehensive plant inspection. This innovative system comprises a robotic platform equipped with a diverse array of sensors integrated to facilitate the detection of various process and infrastructure parameters. These sensors encompass optical (LiDAR, Stereo, UV/IR/RGB cameras), olfactory (electronic nose), and acoustic (microphone array) capabilities, enabling the identification of factors such as methane leaks, flow rates, and infrastructural anomalies. The proposed system underwent individual evaluation at a wastewater treatment site within a chemical plant, providing a practical and challenging environment for testing. The evaluation process encompassed key aspects such as object detection, 3D localization, and path planning. Furthermore, specific evaluations were conducted for optical methane leak detection and localization, as well as acoustic assessments focusing on pump equipment and gas leak localization.

Bayesian optimization has been successfully applied to optimize black-box functions where the number of evaluations is severely limited. However, in many real-world applications, it is hard or impossible to know in advance which designs are feasible due to some physical or system limitations. These issues lead to an even more challenging problem of optimizing an unknown function with unknown constraints. In this paper, we observe that in such scenarios optimal solution typically lies on the boundary between feasible and infeasible regions of the design space, making it considerably more difficult than that with interior optima. Inspired by this observation, we propose BE-CBO, a new Bayesian optimization method that efficiently explores the boundary between feasible and infeasible designs. To identify the boundary, we learn the constraints with an ensemble of neural networks that outperform the standard Gaussian Processes for capturing complex boundaries. Our method demonstrates superior performance against state-of-the-art methods through comprehensive experiments on synthetic and real-world benchmarks.

This research aims to explore the impact of Machine Learning (ML) on the evolution and efficacy of Recommendation Systems (RS), particularly in the context of their growing significance in commercial business environments. Methodologically, the study delves into the role of ML in crafting and refining these systems, focusing on aspects such as data sourcing, feature engineering, and the importance of evaluation metrics, thereby highlighting the iterative nature of enhancing recommendation algorithms. The deployment of Recommendation Engines (RE), driven by advanced algorithms and data analytics, is explored across various domains, showcasing their significant impact on user experience and decision-making processes. These engines not only streamline information discovery and enhance collaboration but also accelerate knowledge acquisition, proving vital in navigating the digital landscape for businesses. They contribute significantly to sales, revenue, and the competitive edge of enterprises by offering improved recommendations that align with individual customer needs. The research identifies the increasing expectation of users for a seamless, intuitive online experience, where content is personalized and dynamically adapted to changing preferences. Future research directions include exploring advancements in deep learning models, ethical considerations in the deployment of RS, and addressing scalability challenges. This study emphasizes the indispensability of comprehending and leveraging ML in RS for researchers and practitioners, to tap into the full potential of personalized recommendation in commercial business prospects.

Research into large-scale crop monitoring has flourished due to increased accessibility to satellite imagery. This review delves into previously unexplored and under-explored areas in sugarcane health monitoring and disease/pest detection using satellite-based spectroscopy and Machine Learning (ML). It discusses key considerations in system development, including relevant satellites, vegetation indices, ML methods, factors influencing sugarcane reflectance, optimal growth conditions, common diseases, and traditional detection methods. Many studies highlight how factors like crop age, soil type, viewing angle, water content, recent weather patterns, and sugarcane variety can impact spectral reflectance, affecting the accuracy of health assessments via spectroscopy. However, these variables have not been fully considered in the literature. In addition, the current literature lacks comprehensive comparisons between ML techniques and vegetation indices. We address these gaps in this review. We discuss that, while current findings suggest the potential for an ML-driven satellite spectroscopy system for monitoring sugarcane health, further research is essential. This paper offers a comprehensive analysis of previous research to aid in unlocking this potential and advancing the development of an effective sugarcane health monitoring system using satellite technology.

Optimizing large-scale wireless networks, including optimal resource management, power allocation, and throughput maximization, is inherently challenging due to their non-observable system dynamics and heterogeneous and complex nature. Herein, a novel ensemble Q-learning algorithm that addresses the performance and complexity challenges of the traditional Q-learning algorithm for optimizing wireless networks is presented. Ensemble learning with synthetic Markov Decision Processes is tailored to wireless networks via new models for approximating large state-space observable wireless networks. In particular, digital cousins are proposed as an extension of the traditional digital twin concept wherein multiple Q-learning algorithms on multiple synthetic Markovian environments are run in parallel and their outputs are fused into a single Q-function. Convergence analyses of key statistics and Q-functions and derivations of upper bounds on the estimation bias and variance are provided. Numerical results across a variety of real-world wireless networks show that the proposed algorithm can achieve up to 50% less average policy error with up to 40% less runtime complexity than the state-of-the-art reinforcement learning algorithms. It is also shown that theoretical results properly predict trends in the experimental results.

Score function estimation is the cornerstone of both training and sampling from diffusion generative models. Despite this fact, the most commonly used estimators are either biased neural network approximations or high variance Monte Carlo estimators based on the conditional score. We introduce a novel nearest neighbour score function estimator which utilizes multiple samples from the training set to dramatically decrease estimator variance. We leverage our low variance estimator in two compelling applications. Training consistency models with our estimator, we report a significant increase in both convergence speed and sample quality. In diffusion models, we show that our estimator can replace a learned network for probability-flow ODE integration, opening promising new avenues of future research.