Generating overtaking trajectories in autonomous racing is a challenging task, as the trajectory must satisfy the vehicle's dynamics and ensure safety and real-time performance running on resource-constrained hardware. This work proposes the Fast and Safe Data-Driven Planner to address this challenge. Sparse Gaussian predictions are introduced to improve both the computational efficiency and accuracy of opponent predictions. Furthermore, the proposed approach employs a bi-level quadratic programming framework to generate an overtaking trajectory leveraging the opponent predictions. The first level uses polynomial fitting to generate a rough trajectory, from which reference states and control inputs are derived for the second level. The second level formulates a model predictive control optimization problem in the Frenet frame, generating a trajectory that satisfies both kinematic feasibility and safety. Experimental results on the F1TENTH platform show that our method outperforms the State-of-the-Art, achieving an 8.93% higher overtaking success rate, allowing the maximum opponent speed, ensuring a smoother ego trajectory, and reducing 74.04% computational time compared to the Predictive Spliner method. The code is available at: https://github.com/ZJU-DDRX/FSDP.

We explore the capability of physics-informed neural networks (PINNs) to discover multiple solutions. Many real-world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty is giving appropriate initial conditions or initial guesses, to which the widely used time-marching schemes and Newton's iteration method are very sensitive in finding solutions for complex computational problems. While machine learning models, particularly PINNs, have shown promise in solving DEs, their ability to capture multiple solutions remains underexplored. In this work, we propose a simple and practical approach using PINNs to learn and discover multiple solutions. We first reveal that PINNs, when combined with random initialization and deep ensemble method -- originally developed for uncertainty quantification -- can effectively uncover multiple solutions to nonlinear ordinary and partial differential equations (ODEs/PDEs). Our approach highlights the critical role of initialization in shaping solution diversity, addressing an often-overlooked aspect of machine learning for scientific computing. Furthermore, we propose utilizing PINN-generated solutions as initial conditions or initial guesses for conventional numerical solvers to enhance accuracy and efficiency in capturing multiple solutions. Extensive numerical experiments, including the Allen-Cahn equation and cavity flow, where our approach successfully identifies both stable and unstable solutions, validate the effectiveness of our method. These findings establish a general and efficient framework for addressing solution multiplicity in nonlinear differential equations.

This paper presents two novel, physics-informed extreme learning machine (PIELM)-based algorithms for solving steady and unsteady nonlinear partial differential equations (PDEs) related to fluid flow. Although single-hidden-layer PIELMs outperform deep physics-informed neural networks (PINNs) in speed and accuracy for linear and quasilinear PDEs, their extension to nonlinear problems remains challenging. To address this, we introduce a curriculum learning strategy that reformulates nonlinear PDEs as a sequence of increasingly complex quasilinear PDEs. Additionally, our approach enables a physically interpretable initialization of network parameters by leveraging Radial Basis Functions (RBFs). The performance of the proposed algorithms is validated on two benchmark incompressible flow problems: the viscous Burgers equation and lid-driven cavity flow. To the best of our knowledge, this is the first work to extend PIELM to solving Burgers' shock solution as well as lid-driven cavity flow up to a Reynolds number of 100. As a practical application, we employ PIELM to predict blood flow in a stenotic vessel. The results confirm that PIELM efficiently handles nonlinear PDEs, positioning it as a promising alternative to PINNs for both linear and nonlinear PDEs.

MAV-capturing-MAV (MCM) is one of the few effective methods for physically countering misused or malicious MAVs.This paper presents a vision-based cooperative MCM system, where multiple pursuer MAVs equipped with onboard vision systems detect, localize, and pursue a target MAV. To enhance robustness, a distributed state estimation and control framework enables the pursuer MAVs to autonomously coordinate their actions. Pursuer trajectories are optimized using Model Predictive Control (MPC) and executed via a low-level SO(3) controller, ensuring smooth and stable pursuit. Once the capture conditions are satisfied, the pursuer MAVs automatically deploy a flying net to intercept the target. These capture conditions are determined based on the predicted motion of the net. To enable real-time decision-making, we propose a lightweight computational method to approximate the net motion, avoiding the prohibitive cost of solving the full net dynamics. The effectiveness of the proposed system is validated through simulations and real-world experiments. In real-world tests, our approach successfully captures a moving target traveling at 4 meters per second with an acceleration of 1 meter per square second, achieving a success rate of 64.7 percent.

This paper deals with the identification of the stochastic Ornstein-Uhlenbeck (OU) process error model, which is characterized by an inverse time constant, and the unknown variances of the process and observation noises. Although the availability of the explicit expression of the log-likelihood function allows one to obtain the maximum likelihood estimator (MLE), this entails evaluating the nontrivial gradient and also often struggles with local optima. To address these limitations, we put forth a sample-efficient global optimization approach based on the Bayesian optimization (BO) framework, which relies on a Gaussian process (GP) surrogate model for the objective function that effectively balances exploration and exploitation to select the query points. Specifically, each evaluation of the objective is implemented efficiently through the Kalman filter (KF) recursion. Comprehensive experiments on various parameter settings and sampling intervals corroborate that BO-based estimator consistently outperforms MLE implemented by the steady-state KF approximation and the expectation-maximization algorithm (whose derivation is a side contribution) in terms of root mean-square error (RMSE) and statistical consistency, confirming the effectiveness and robustness of the BO for identification of the stochastic OU process. Notably, the RMSE values produced by the BO-based estimator are smaller than the classical Cram\'{e}r-Rao lower bound, especially for the inverse time constant, estimating which has been a long-standing challenge. This seemingly counterintuitive result can be explained by the data-driven prior for the learning parameters indirectly injected by BO through the GP prior over the objective function.

With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics at the high-school, undergraduate, and graduate level, there are far fewer resources available that align with the level of difficulty and open endedness encountered by professional mathematicians working on open problems. To address this, we introduce a new collection of datasets, the Algebraic Combinatorics Dataset Repository (ACD Repo), representing either foundational results or open problems in algebraic combinatorics, a subfield of mathematics that studies discrete structures arising from abstract algebra. Further differentiating our dataset collection is the fact that it aims at the conjecturing process. Each dataset includes an open-ended research-level question and a large collection of examples (up to 10M in some cases) from which conjectures should be generated. We describe all nine datasets, the different ways machine learning models can be applied to them (e.g., training with narrow models followed by interpretability analysis or program synthesis with LLMs), and discuss some of the challenges involved in designing datasets like these.

In the rapidly evolving digital era, comprehending the intricate dynamics influencing server power consumption, efficiency, and performance is crucial for sustainable data center operations. However, existing models lack the ability to provide a detailed and reliable understanding of these intricate relationships. This study employs a machine learning-based approach, using the SPECPower_ssj2008 database, to facilitate user-friendly and generalizable server modeling. The resulting models demonstrate high accuracy, with errors falling within approximately 10% on the testing dataset, showcasing their practical utility and generalizability. Through meticulous analysis, predictive features related to hardware availability date, server workload level, and specifications are identified, providing insights into optimizing energy conservation, efficiency, and performance in server deployment and operation. By systematically measuring biases and uncertainties, the study underscores the need for caution when employing historical data for prospective server modeling, considering the dynamic nature of technology landscapes. Collectively, this work offers valuable insights into the sustainable deployment and operation of servers in data centers, paving the way for enhanced resource use efficiency and more environmentally conscious practices.

To adapt to real-world data streams, continual learning (CL) systems must rapidly learn new concepts while preserving and utilizing prior knowledge. When it comes to adding new information to continually-trained deep neural networks (DNNs), classifier weights for newly encountered categories are typically initialized randomly, leading to high initial training loss (spikes) and instability. Consequently, achieving optimal convergence and accuracy requires prolonged training, increasing computational costs. Inspired by Neural Collapse (NC), we propose a weight initialization strategy to improve learning efficiency in CL. In DNNs trained with mean-squared-error, NC gives rise to a Least-Square (LS) classifier in the last layer, whose weights can be analytically derived from learned features. Our method mitigates initial loss spikes and accelerates adaptation to new tasks. We evaluate our approach in large-scale CL settings, demonstrating faster adaptation and improved CL performance. Deep learning models excel in static environments where the data follows an independent and identically distributed (IID) assumption. However, in real-world scenarios, data distributions shift over time (non-IID), and new data arrives sequentially. Conventional deep neural networks (DNNs) struggle under such conditions, often requiring periodic re-training from scratch, which is not only computationally expensive but also contributes significantly to the carbon footprint of AI (Schwartz et al., 2020). Despite frequent retraining from scratch, real-world models still suffer up to 40% accuracy drops (Mallick et al., 2022). Continual learning (CL) aims to address this inefficiency by enabling models to learn from evolving data streams while preserving previously acquired knowledge (Parisi et al., 2019). CL is a promising solution to model decay, where predictive performance deteriorates over time due to concept drift--a shift in the meaning or distribution of target variables (Tsymbal, 2004; Gama et al., 2014; Lu et al., 2018).

Node embedding is a key technique for representing graph nodes as vectors while preserving structural and relational properties, which enables machine learning tasks like feature extraction, clustering, and classification. While classical methods such as DeepWalk, node2vec, and graph convolutional networks learn node embeddings by capturing structural and relational patterns in graphs, they often require significant computational resources and struggle with scalability on large graphs. Quantum computing provides a promising alternative for graph-based learning by leveraging quantum effects and introducing novel optimization approaches. Variational quantum circuits and quantum kernel methods have been explored for embedding tasks, but their scalability remains limited due to the constraints of noisy intermediate-scale quantum (NISQ) hardware. In this paper, we investigate quantum annealing (QA) as an alternative approach that mitigates key challenges associated with quantum gate-based models. We propose several formulations of the node embedding problem as a quadratic unconstrained binary optimization (QUBO) instance, making it compatible with current quantum annealers such as those developed by D-Wave. We implement our algorithms on a D-Wave quantum annealer and evaluate their performance on graphs with up to 100 nodes and embedding dimensions of up to 5. Our findings indicate that QA is a viable approach for graph-based learning, providing a scalable and efficient alternative to previous quantum embedding techniques.

Using the information theory, this study provides insights into how the construction of latent space of autoencoder (AE) using deep neural network (DNN) training finds a smooth low-dimensional manifold in the stiff dynamical system. Our recent study [1] reported that an autoencoder (AE) combined with neural ODE (NODE) as a surrogate reduced order model (ROM) for the integration of stiff chemically reacting systems led to a significant reduction in the temporal stiffness, and the behavior was attributed to the identification of a slow invariant manifold by the nonlinear projection of the AE. The present work offers fundamental understanding of the mechanism by employing concepts from information theory and better mixing. The learning mechanism of both the encoder and decoder are explained by plotting the evolution of mutual information and identifying two different phases. Subsequently, the density distribution is plotted for the physical and latent variables, which shows the transformation of the \emph{rare event} in the physical space to a \emph{highly likely} (more probable) event in the latent space provided by the nonlinear autoencoder. Finally, the nonlinear transformation leading to density redistribution is explained using concepts from information theory and probability.