In this paper, we study the stochastic collocation (SC) methods for uncertainty quantification (UQ) in hyperbolic systems of nonlinear partial differential equations (PDEs). In these methods, the underlying PDEs are numerically solved at a set of collocation points in random space. A standard SC approach is based on a generalized polynomial chaos (gPC) expansion, which relies on choosing the collocation points based on the prescribed probability distribution and approximating the computed solution by a linear combination of orthogonal polynomials in the random variable. We demonstrate that this approach struggles to accurately capture discontinuous solutions, often leading to oscillations (Gibbs phenomenon) that deviate significantly from the physical solutions. We explore alternative SC methods, in which one can choose an arbitrary set of collocation points and employ shape-preserving splines to interpolate the solution in a random space. Our study demonstrates the effectiveness of spline-based collocation in accurately capturing and assessing uncertainties while suppressing oscillations. We illustrate the superiority of the spline-based collocation on two numerical examples, including the inviscid Burgers and shallow water equations.
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In this work we investigate an inverse coefficient problem for the one-dimensional subdiffusion model, which involves a Caputo fractional derivative in time. The inverse problem is to determine two coefficients and multiple parameters (the order, and length of the interval) from one pair of lateral Cauchy data. The lateral Cauchy data are given on disjoint sets in time with a single excitation and the measurement is made on a time sequence located outside the support of the excitation. We prove two uniqueness results for different lateral Cauchy data. The analysis is based on the solution representation, analyticity of the observation and a refined version of inverse Sturm-Liouville theory due to Sini [35]. Our results heavily exploit the memory effect of fractional diffusion for the unique recovery of the coefficients in the model. Several numerical experiments are also presented to complement the analysis.
In this study, we introduce a novel framework called Toast for learning general-purpose representations of road networks, along with its advanced counterpart DyToast, designed to enhance the integration of temporal dynamics to boost the performance of various time-sensitive downstream tasks. Specifically, we propose to encode two pivotal semantic characteristics intrinsic to road networks: traffic patterns and traveling semantics. To achieve this, we refine the skip-gram module by incorporating auxiliary objectives aimed at predicting the traffic context associated with a target road segment. Moreover, we leverage trajectory data and design pre-training strategies based on Transformer to distill traveling semantics on road networks. DyToast further augments this framework by employing unified trigonometric functions characterized by their beneficial properties, enabling the capture of temporal evolution and dynamic nature of road networks more effectively. With these proposed techniques, we can obtain representations that encode multi-faceted aspects of knowledge within road networks, applicable across both road segment-based applications and trajectory-based applications. Extensive experiments on two real-world datasets across three tasks demonstrate that our proposed framework consistently outperforms the state-of-the-art baselines by a significant margin.
In this work, we study the out-of-distribution (OOD) detection problem through the use of the feature space of a pre-trained deep classifier. We show that learning the density of in-distribution (ID) features with an energy-based models (EBM) leads to competitive detection results. However, we found that the non-mixing of MCMC sampling during the EBM's training undermines its detection performance. To overcome this an energy-based correction of a mixture of class-conditional Gaussian distributions. We obtains favorable results when compared to a strong baseline like the KNN detector on the CIFAR-10/CIFAR-100 OOD detection benchmarks.
In this paper, we study the individual preference (IP) stability, which is an notion capturing individual fairness and stability in clustering. Within this setting, a clustering is $\alpha$-IP stable when each data point's average distance to its cluster is no more than $\alpha$ times its average distance to any other cluster. In this paper, we study the natural local search algorithm for IP stable clustering. Our analysis confirms a $O(\log n)$-IP stability guarantee for this algorithm, where $n$ denotes the number of points in the input. Furthermore, by refining the local search approach, we show it runs in an almost linear time, $\tilde{O}(nk)$.
In turbulence modeling, we are concerned with finding closure models that represent the effect of the subgrid scales on the resolved scales. Recent approaches gravitate towards machine learning techniques to construct such models. However, the stability of machine-learned closure models and their abidance by physical structure (e.g. symmetries, conservation laws) are still open problems. To tackle both issues, we take the `discretize first, filter next' approach. In this approach we apply a spatial averaging filter to existing fine-grid discretizations. The main novelty is that we introduce an additional set of equations which dynamically model the energy of the subgrid scales. Having an estimate of the energy of the subgrid scales, we can use the concept of energy conservation to derive stability. The subgrid energy containing variables are determined via a data-driven technique. The closure model is used to model the interaction between the filtered quantities and the subgrid energy. Therefore the total energy should be conserved. Abiding by this conservation law yields guaranteed stability of the system. In this work, we propose a novel skew-symmetric convolutional neural network architecture that satisfies this law. The result is that stability is guaranteed, independent of the weights and biases of the network. Importantly, as our framework allows for energy exchange between resolved and subgrid scales it can model backscatter. To model dissipative systems (e.g. viscous flows), the framework is extended with a diffusive component. The introduced neural network architecture is constructed such that it also satisfies momentum conservation. We apply the new methodology to both the viscous Burgers' equation and the Korteweg-De Vries equation in 1D. The novel architecture displays superior stability properties when compared to a vanilla convolutional neural network.
Graph Enhanced Reinforcement Learning for Effective Group Formation in Collaborative Problem Solving
This study addresses the challenge of forming effective groups in collaborative problem-solving environments. Recognizing the complexity of human interactions and the necessity for efficient collaboration, we propose a novel approach leveraging graph theory and reinforcement learning. Our methodology involves constructing a graph from a dataset where nodes represent participants, and edges signify the interactions between them. We conceptualize each participant as an agent within a reinforcement learning framework, aiming to learn an optimal graph structure that reflects effective group dynamics. Clustering techniques are employed to delineate clear group structures based on the learned graph. Our approach provides theoretical solutions based on evaluation metrics and graph measurements, offering insights into potential improvements in group effectiveness and reductions in conflict incidences. This research contributes to the fields of collaborative work and educational psychology by presenting a data-driven, analytical approach to group formation. It has practical implications for organizational team building, classroom settings, and any collaborative scenario where group dynamics are crucial. The study opens new avenues for exploring the application of graph theory and reinforcement learning in social and behavioral sciences, highlighting the potential for empirical validation in future work.
In this paper, we introduce an improved approach of speculative decoding aimed at enhancing the efficiency of serving large language models. Our method capitalizes on the strengths of two established techniques: the classic two-model speculative decoding approach, and the more recent single-model approach, Medusa. Drawing inspiration from Medusa, our approach adopts a single-model strategy for speculative decoding. However, our method distinguishes itself by employing a single, lightweight draft head with a recurrent dependency design, akin in essence to the small, draft model uses in classic speculative decoding, but without the complexities of the full transformer architecture. And because of the recurrent dependency, we can use beam search to swiftly filter out undesired candidates with the draft head. The outcome is a method that combines the simplicity of single-model design and avoids the need to create a data-dependent tree attention structure only for inference in Medusa. We empirically demonstrate the effectiveness of the proposed method on several popular open source language models, along with a comprehensive analysis of the trade-offs involved in adopting this approach.
In this paper, we propose a model predictive control (MPC) that accomplishes interactive robotic tasks, in which multiple contacts may occur at unknown locations. To address such scenarios, we made an explicit contact feedback loop in the MPC framework. An algorithm called Multi-Contact Particle Filter with Exploration Particle (MCP-EP) is employed to establish real-time feedback of multi-contact information. Then the interaction locations and forces are accommodated in the MPC framework via a spring contact model. Moreover, we achieved real-time control for a 7 degrees of freedom robot without any simplifying assumptions by employing a Differential-Dynamic-Programming algorithm. We achieved 6.8kHz, 1.9kHz, and 1.8kHz update rates of the MPC for 0, 1, and 2 contacts, respectively. This allows the robot to handle unexpected contacts in real time. Real-world experiments show the effectiveness of the proposed method in various scenarios.
In this paper, we address the problem of estimating the rotational extrinsics, as well as the scale factors of two gyroscopes rigidly mounted on the same device. In particular, we formulate the problem as a least-squares minimization and introduce a direct algorithm that computes the estimated quantities without any iterations, hence avoiding local minima and improving efficiency. Furthermore, we show that the rotational extrinsics are observable while the scale factors can be determined up to global scale for general configurations of the gyroscopes. To this end, we also study special placements of the gyroscopes where a pair, or all, of their axes are parallel and analyze their impact on the scale factors' observability. Lastly, we evaluate our algorithm in simulations and real-world experiments to assess its performance as a function of key motion and sensor characteristics.
We describe ACE0, a lightweight platform for evaluating the suitability and viability of AI methods for behaviour discovery in multiagent simulations. Specifically, ACE0 was designed to explore AI methods for multi-agent simulations used in operations research studies related to new technologies such as autonomous aircraft. Simulation environments used in production are often high-fidelity, complex, require significant domain knowledge and as a result have high R&D costs. Minimal and lightweight simulation environments can help researchers and engineers evaluate the viability of new AI technologies for behaviour discovery in a more agile and potentially cost effective manner. In this paper we describe the motivation for the development of ACE0.We provide a technical overview of the system architecture, describe a case study of behaviour discovery in the aerospace domain, and provide a qualitative evaluation of the system. The evaluation includes a brief description of collaborative research projects with academic partners, exploring different AI behaviour discovery methods.
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