Follow-the-Regularized-Leader with Adversarial Constraints

Constrained Online Convex Optimization (COCO) can be seen as a generalization of the standard Online Convex Optimization (OCO) framework. At each round, a cost function and constraint function are revealed after a learner chooses an action. The goal is to minimize both the regret and cumulative constraint violation (CCV) against an adaptive adversary. We show for the first time that is possible to obtain the optimal $O(\sqrt{T})$ bound on both regret and CCV, improving the best known bounds of $O \left( \sqrt{T} \right)$ and $\~{O} \left( \sqrt{T} \right)$ for the regret and CCV, respectively.

Advancing Solutions for the Three-Body Problem Through Physics-Informed Neural Networks

First formulated by Sir Isaac Newton in his work "Philosophiae Naturalis Principia Mathematica", the concept of the Three-Body Problem was put forth as a study of the motion of the three celestial bodies within the Earth-Sun-Moon system. In a generalized definition, it seeks to predict the motion for an isolated system composed of three point masses freely interacting under Newton's law of universal attraction. This proves to be analogous to a multitude of interactions between celestial bodies, and thus, the problem finds applicability within the studies of celestial mechanics. Despite numerous attempts by renowned physicists to solve it throughout the last three centuries, no general closed-form solutions have been reached due to its inherently chaotic nature for most initial conditions. Current state-of-the-art solutions are based on two approaches, either numerical high-precision integration or machine learning-based. Notwithstanding the breakthroughs of neural networks, these present a significant limitation, which is their ignorance of any prior knowledge of the chaotic systems presented. Thus, in this work, we propose a novel method that utilizes Physics-Informed Neural Networks (PINNs). These deep neural networks are able to incorporate any prior system knowledge expressible as an Ordinary Differential Equation (ODE) into their learning processes as a regularizing agent. Our findings showcase that PINNs surpass current state-of-the-art machine learning methods with comparable prediction quality. Despite a better prediction quality, the usability of numerical integrators suffers due to their prohibitively high computational cost. These findings confirm that PINNs are both effective and time-efficient open-form solvers of the Three-Body Problem that capitalize on the extensive knowledge we hold of classical mechanics.

Generalizing Trilateration: Approximate Maximum Likelihood Estimator for Initial Orbit Determination in Low-Earth Orbit

With the increase in the number of active satellites and space debris in orbit, the problem of initial orbit determination (IOD) becomes increasingly important, demanding a high accuracy. Over the years, different approaches have been presented such as filtering methods (for example, Extended Kalman Filter), differential algebra or solving Lambert's problem. In this work, we consider a setting of three monostatic radars, where all available measurements are taken approximately at the same instant. This follows a similar setting as trilateration, a state-of-the-art approach, where each radar is able to obtain a single measurement of range and range-rate. Differently, and due to advances in Multiple-Input Multiple-Output (MIMO) radars, we assume that each location is able to obtain a larger set of range, angle and Doppler shift measurements. Thus, our method can be understood as an extension of trilateration leveraging more recent technology and incorporating additional data. We formulate the problem as a Maximum Likelihood Estimator (MLE), which for some number of observations is asymptotically unbiased and asymptotically efficient. Through numerical experiments, we demonstrate that our method attains the same accuracy as the trilateration method for the same number of measurements and offers an alternative and generalization, returning a more accurate estimation of the satellite's state vector, as the number of available measurements increases.

TS40K: a 3D Point Cloud Dataset of Rural Terrain and Electrical Transmission System

Research on supervised learning algorithms in 3D scene understanding has risen in prominence and witness great increases in performance across several datasets. The leading force of this research is the problem of autonomous driving followed by indoor scene segmentation. However, openly available 3D data on these tasks mainly focuses on urban scenarios. In this paper, we propose TS40K, a 3D point cloud dataset that encompasses more than 40,000 Km on electrical transmission systems situated in European rural terrain. This is not only a novel problem for the research community that can aid in the high-risk mission of power-grid inspection, but it also offers 3D point clouds with distinct characteristics from those in self-driving and indoor 3D data, such as high point-density and no occlusion. In our dataset, each 3D point is labeled with 1 out of 22 annotated classes. We evaluate the performance of state-of-the-art methods on our dataset concerning 3D semantic segmentation and 3D object detection. Finally, we provide a comprehensive analysis of the results along with key challenges such as using labels that were not originally intended for learning tasks.

PeersimGym: An Environment for Solving the Task Offloading Problem with Reinforcement Learning

Task offloading, crucial for balancing computational loads across devices in networks such as the Internet of Things, poses significant optimization challenges, including minimizing latency and energy usage under strict communication and storage constraints. While traditional optimization falls short in scalability; and heuristic approaches lack in achieving optimal outcomes, Reinforcement Learning (RL) offers a promising avenue by enabling the learning of optimal offloading strategies through iterative interactions. However, the efficacy of RL hinges on access to rich datasets and custom-tailored, realistic training environments. To address this, we introduce PeersimGym, an open-source, customizable simulation environment tailored for developing and optimizing task offloading strategies within computational networks. PeersimGym supports a wide range of network topologies and computational constraints and integrates a \textit{PettingZoo}-based interface for RL agent deployment in both solo and multi-agent setups. Furthermore, we demonstrate the utility of the environment through experiments with Deep Reinforcement Learning agents, showcasing the potential of RL-based approaches to significantly enhance offloading strategies in distributed computing settings. PeersimGym thus bridges the gap between theoretical RL models and their practical applications, paving the way for advancements in efficient task offloading methodologies.

One-Shot Initial Orbit Determination in Low-Earth Orbit

Due to the importance of satellites for society and the exponential increase in the number of objects in orbit, it is important to accurately determine the state (e.g., position and velocity) of these Resident Space Objects (RSOs) at any time and in a timely manner. State-of-the-art methodologies for initial orbit determination consist of Kalman-type filters that process sequential data over time and return the state and associated uncertainty of the object, as is the case of the Extended Kalman Filter (EKF). However, these methodologies are dependent on a good initial guess for the state vector and usually simplify the physical dynamical model, due to the difficulty of precisely modeling perturbative forces, such as atmospheric drag and solar radiation pressure. Other approaches do not require assumptions about the dynamical system, such as the trilateration method, and require simultaneous measurements, such as three measurements of range and range-rate for the particular case of trilateration. We consider the same setting of simultaneous measurements (one-shot), resorting to time delay and Doppler shift measurements. Based on recent advancements in the problem of moving target localization for sonar multistatic systems, we are able to formulate the problem of initial orbit determination as a Weighted Least Squares. With this approach, we are able to directly obtain the state of the object (position and velocity) and the associated covariance matrix from the Fisher's Information Matrix (FIM). We demonstrate that, for small noise, our estimator is able to attain the Cram\'er-Rao Lower Bound accuracy, i.e., the accuracy attained by the unbiased estimator with minimum variance. We also numerically demonstrate that our estimator is able to attain better accuracy on the state estimation than the trilateration method and returns a smaller uncertainty associated with the estimation.

Predicting the Probability of Collision of a Satellite with Space Debris: A Bayesian Machine Learning Approach

Space is becoming more crowded in Low Earth Orbit due to increased space activity. Such a dense space environment increases the risk of collisions between space objects endangering the whole space population. Therefore, the need to consider collision avoidance as part of routine operations is evident to satellite operators. Current procedures rely on the analysis of multiple collision warnings by human analysts. However, with the continuous growth of the space population, this manual approach may become unfeasible, highlighting the importance of automation in risk assessment. In 2019, ESA launched a competition to study the feasibility of applying machine learning in collision risk estimation and released a dataset that contained sequences of Conjunction Data Messages (CDMs) in support of real close encounters. The competition results showed that the naive forecast and its variants are strong predictors for this problem, which suggests that the CDMs may follow the Markov property. The proposed work investigates this theory by benchmarking Hidden Markov Models (HMM) in predicting the risk of collision between two resident space objects by using one feature of the entire dataset: the sequence of the probability in the CDMs. In addition, Bayesian statistics are used to infer a joint distribution for the parameters of the models, which allows the development of robust and reliable probabilistic predictive models that can incorporate physical or prior knowledge about the problem within a rigorous theoretical framework and provides prediction uncertainties that nicely reflect the accuracy of the predicted risk. This work shows that the implemented HMM outperforms the naive solution in some metrics, which further adds to the idea that the collision warnings may be Markovian and suggests that this is a powerful method to be further explored.

Finding Real-World Orbital Motion Laws from Data

A novel approach is presented for discovering PDEs that govern the motion of satellites in space. The method is based on SINDy, a data-driven technique capable of identifying the underlying dynamics of complex physical systems from time series data. SINDy is utilized to uncover PDEs that describe the laws of physics in space, which are non-deterministic and influenced by various factors such as drag or the reference area (related to the attitude of the satellite). In contrast to prior works, the physically interpretable coordinate system is maintained, and no dimensionality reduction technique is applied to the data. By training the model with multiple representative trajectories of LEO - encompassing various inclinations, eccentricities, and altitudes - and testing it with unseen orbital motion patterns, a mean error of around 140 km for the positions and 0.12 km/s for the velocities is achieved. The method offers the advantage of delivering interpretable, accurate, and complex models of orbital motion that can be employed for propagation or as inputs to predictive models for other variables of interest, such as atmospheric drag or the probability of collision in an encounter with a spacecraft or space objects. In conclusion, the work demonstrates the promising potential of using SINDy to discover the equations governing the behaviour of satellites in space. The technique has been successfully applied to uncover PDEs describing the motion of satellites in LEO with high accuracy. The method possesses several advantages over traditional models, including the ability to provide physically interpretable, accurate, and complex models of orbital motion derived from high-entropy datasets. These models can be utilised for propagation or as inputs to predictive models for other variables of interest.

Probability of Collision of satellites and space debris for short-term encounters: Rederivation and fast-to-compute upper and lower bounds

The proliferation of space debris in LEO has become a major concern for the space industry. With the growing interest in space exploration, the prediction of potential collisions between objects in orbit has become a crucial issue. It is estimated that, in orbit, there are millions of fragments a few millimeters in size and thousands of inoperative satellites and discarded rocket stages. Given the high speeds that these fragments can reach, even fragments a few millimeters in size can cause fractures in a satellite's hull or put a serious crack in the window of a space shuttle. The conventional method proposed by Akella and Alfriend in 2000 remains widely used to estimate the probability of collision in short-term encounters. Given the small period of time, it is assumed that, during the encounter: (1) trajectories are represented by straight lines with constant velocity; (2) there is no velocity uncertainty and the position exhibits a stationary distribution throughout the encounter; and (3) position uncertainties are independent and represented by Gaussian distributions. This study introduces a novel derivation based on first principles that naturally allows for tight and fast upper and lower bounds for the probability of collision. We tested implementations of both probability and bound computations with the original and our formulation on a real CDM dataset used in ESA's Collision Avoidance Challenge. Our approach reduces the calculation of the probability to two one-dimensional integrals and has the potential to significantly reduce the processing time compared to the traditional method, from 80% to nearly real-time.

Statistical Learning of Conjunction Data Messages Through a Bayesian Non-Homogeneous Poisson Process

Current approaches for collision avoidance and space traffic management face many challenges, mainly due to the continuous increase in the number of objects in orbit and the lack of scalable and automated solutions. To avoid catastrophic incidents, satellite owners/operators must be aware of their assets' collision risk to decide whether a collision avoidance manoeuvre needs to be performed. This process is typically executed through the use of warnings issued in the form of CDMs which contain information about the event, such as the expected TCA and the probability of collision. Our previous work presented a statistical learning model that allowed us to answer two important questions: (1) Will any new conjunctions be issued in the next specified time interval? (2) When and with what uncertainty will the next CDM arrive? However, the model was based on an empirical Bayes homogeneous Poisson process, which assumes that the arrival rates of CDMs are constant over time. In fact, the rate at which the CDMs are issued depends on the behaviour of the objects as well as on the screening process performed by third parties. Thus, in this work, we extend the previous study and propose a Bayesian non-homogeneous Poisson process implemented with high precision using a Probabilistic Programming Language to fully describe the underlying phenomena. We compare the proposed solution with a baseline model to demonstrate the added value of our approach. The results show that this problem can be successfully modelled by our Bayesian non-homogeneous Poisson Process with greater accuracy, contributing to the development of automated collision avoidance systems and helping operators react timely but sparingly with satellite manoeuvres.