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.

The effectiveness of instruction fine-tuning for Large Language Models is fundamentally constrained by the quality and efficiency of training datasets. This work introduces Low-Confidence Gold (LCG), a novel filtering framework that employs centroid-based clustering and confidence-guided selection for identifying valuable instruction pairs. Through a semi-supervised approach using a lightweight classifier trained on representative samples, LCG curates high-quality subsets while preserving data diversity. Experimental evaluation demonstrates that models fine-tuned on LCG-filtered subsets of 6K samples achieve superior performance compared to existing methods, with substantial improvements on MT-bench and consistent gains across comprehensive evaluation metrics. The framework's efficacy while maintaining model performance establishes a promising direction for efficient instruction tuning.

Recent research on instant runoff voting (IRV) shows that it exhibits a striking combinatorial property in one-dimensional preference spaces: there is an "exclusion zone" around the median voter such that if a candidate from the exclusion zone is on the ballot, then the winner must come from the exclusion zone. Thus, in one dimension, IRV cannot elect an extreme candidate as long as a sufficiently moderate candidate is running. In this work, we examine the mathematical structure of exclusion zones as a broad phenomenon in more general preference spaces. We prove that with voters uniformly distributed over any $d$-dimensional hyperrectangle (for $d > 1$), IRV has no nontrivial exclusion zone. However, we also show that IRV exclusion zones are not solely a one-dimensional phenomenon. For irregular higher-dimensional preference spaces with fewer symmetries than hyperrectangles, IRV can exhibit nontrivial exclusion zones. As a further exploration, we study IRV exclusion zones in graph voting, where nodes represent voters who prefer candidates closer to them in the graph. Here, we show that IRV exclusion zones present a surprising computational challenge: even checking whether a given set of positions is an IRV exclusion zone is NP-hard. We develop an efficient randomized approximation algorithm for checking and finding exclusion zones. We also report on computational experiments with exclusion zones in two directions: (i) applying our approximation algorithm to a collection of real-world school friendship networks, we find that about 60% of these networks have probable nontrivial IRV exclusion zones; and (ii) performing an exhaustive computer search of small graphs and trees, we also find nontrivial IRV exclusion zones in most graphs. While our focus is on IRV, the properties of exclusion zones we establish provide a novel method for analyzing voting systems in metric spaces more generally.

Accurate lifetime prediction of structures subjected to cyclic loading is vital, especially in scenarios involving non-uniform loading histories where load sequencing critically influences structural durability. Addressing this complexity requires advanced modeling approaches capable of capturing the intricate relationship between loading sequences and fatigue lifetime. Traditional fatigue simulations are computationally prohibitive, necessitating more efficient methods. This study highlights the potential of physics-based machine learning ($\phi$ML) to predict the fatigue lifetime of materials. Specifically, a FFNN is designed to embed physical constraints from experimental evidence directly into its architecture to enhance prediction accuracy. It is trained using numerical simulations generated by a physically based anisotropic continuum damage fatigue model. The model is calibrated and validated against experimental fatigue data of concrete cylinder specimens tested in uniaxial compression. The proposed approach demonstrates superior accuracy compared to purely data-driven neural networks, particularly in situations with limited training data, achieving realistic predictions of damage accumulation. Thus, a general algorithm is developed and successfully applied to predict fatigue lifetimes under complex loading scenarios with multiple loading ranges. Hereby, the $\phi$ML model serves as a surrogate to capture damage evolution across load transitions. The $\phi$ML based algorithm is subsequently employed to investigate the influence of multiple loading transitions on accumulated fatigue life, and its predictions align with trends observed in recent experimental studies. This work demonstrates $\phi$ML as a promising technique for efficient and reliable fatigue life prediction in engineering structures, with possible integration into digital twin models for real-time assessment.

We perform Bayesian optimization using a Gaussian process perspective on Bayesian Additive Regression Trees (BART). Our BART Kernel (BARK) uses tree agreement to define a posterior over piecewise-constant functions, and we explore the space of tree kernels using a Markov chain Monte Carlo approach. Where BART only samples functions, the resulting BARK model obtains samples of Gaussian processes defining distributions over functions, which allow us to build acquisition functions for Bayesian optimization. Our tree-based approach enables global optimization over the surrogate, even for mixed-feature spaces. Moreover, where many previous tree-based kernels provide uncertainty quantification over function values, our sampling scheme captures uncertainty over the tree structure itself. Our experiments show the strong performance of BARK on both synthetic and applied benchmarks, due to the combination of our fully Bayesian surrogate and the optimization procedure.

Highly accurate force fields are a mandatory requirement to generate predictive simulations. In this regard, Machine Learning Force Fields (MLFFs) have emerged as a revolutionary approach in computational chemistry and materials science, combining the accuracy of quantum mechanical methods with computational efficiency orders of magnitude superior to ab-initio methods. This chapter provides an introduction of the fundamentals of learning and how it is applied to construct MLFFs, detailing key methodologies such as neural network potentials and kernel-based models. Emphasis is placed on the construction of SchNet model, as one of the most elemental neural network-based force fields that are nowadays the basis of modern architectures. Additionally, the GDML framework is described in detail as an example of how the elegant formulation of kernel methods can be used to construct mathematically robust and physics-inspired MLFFs. The ongoing advancements in MLFF development continue to expand their applicability, enabling precise simulations of large and complex systems that were previously beyond reach. This chapter concludes by highlighting the transformative impact of MLFFs on scientific research, underscoring their role in driving future discoveries in the fields of chemistry, physics, and materials science.

Large Language Models (LLMs) are rapidly saturating existing benchmarks, necessitating new open-ended evaluations. We introduce the Factorio Learning Environment (FLE), based on the game of Factorio, that tests agents in long-term planning, program synthesis, and resource optimization. FLE provides exponentially scaling challenges -- from basic automation to complex factories processing millions of resource units per second. We provide two settings: (1) lab-play consisting of eight structured tasks with fixed resources, and (2) open-play with the unbounded task of building the largest factory on an procedurally generated map. We demonstrate across both settings that models still lack strong spatial reasoning. In lab-play, we find that LLMs exhibit promising short-horizon skills, yet are unable to operate effectively in constrained environments, reflecting limitations in error analysis. In open-play, while LLMs discover automation strategies that improve growth (e.g electric-powered drilling), they fail to achieve complex automation (e.g electronic-circuit manufacturing).

The emerging Internet of Things (IoT) applications, such as driverless cars, have a growing demand for high-precision positioning and navigation. Nowadays, LiDAR inertial odometry becomes increasingly prevalent in robotics and autonomous driving. However, many current SLAM systems lack sufficient adaptability to various scenarios. Challenges include decreased point cloud accuracy with longer frame intervals under the constant velocity assumption, coupling of erroneous IMU information when IMU saturation occurs, and decreased localization accuracy due to the use of fixed-resolution maps during indoor-outdoor scene transitions. To address these issues, we propose a loosely coupled adaptive LiDAR-Inertial-Odometry named \textbf{Adaptive-LIO}, which incorporates adaptive segmentation to enhance mapping accuracy, adapts motion modality through IMU saturation and fault detection, and adjusts map resolution adaptively using multi-resolution voxel maps based on the distance from the LiDAR center. Our proposed method has been tested in various challenging scenarios, demonstrating the effectiveness of the improvements we introduce. The code is open-source on GitHub: \href{https://github.com/chengwei0427/adaptive_lio}{Adaptive-LIO}.

The rise of Large Language Models (LLMs) and generative visual analytics systems has transformed data-driven insights, yet significant challenges persist in accurately interpreting users' analytical and interaction intents. While language inputs offer flexibility, they often lack precision, making the expression of complex intents inefficient, error-prone, and time-intensive. To address these limitations, we investigate the design space of multimodal interactions for generative visual analytics through a literature review and pilot brainstorming sessions. Building on these insights, we introduce a highly extensible workflow that integrates multiple LLM agents for intent inference and visualization generation. We develop InterChat, a generative visual analytics system that combines direct manipulation of visual elements with natural language inputs. This integration enables precise intent communication and supports progressive, visually driven exploratory data analyses. By employing effective prompt engineering, and contextual interaction linking, alongside intuitive visualization and interaction designs, InterChat bridges the gap between user interactions and LLM-driven visualizations, enhancing both interpretability and usability. Extensive evaluations, including two usage scenarios, a user study, and expert feedback, demonstrate the effectiveness of InterChat. Results show significant improvements in the accuracy and efficiency of handling complex visual analytics tasks, highlighting the potential of multimodal interactions to redefine user engagement and analytical depth in generative visual analytics.

We explore neuro-symbolic approaches to generalize actionable knowledge, enabling embodied agents to tackle complex tasks more effectively in open-domain environments. A key challenge for embodied agents is the generalization of knowledge across diverse environments and situations, as limited experiences often confine them to their prior knowledge. To address this issue, we introduce a novel framework, NeSyC, a neuro-symbolic continual learner that emulates the hypothetico-deductive model by continually formulating and validating knowledge from limited experiences through the combined use of Large Language Models (LLMs) and symbolic tools. Specifically, we devise a contrastive generality improvement scheme within NeSyC, which iteratively generates hypotheses using LLMs and conducts contrastive validation via symbolic tools. This scheme reinforces the justification for admissible actions while minimizing the inference of inadmissible ones. Additionally, we incorporate a memory-based monitoring scheme that efficiently detects action errors and triggers the knowledge refinement process across domains. Experiments conducted on diverse embodied task benchmarks-including ALFWorld, VirtualHome, Minecraft, RLBench, and a real-world robotic scenario-demonstrate that NeSyC is highly effective in solving complex embodied tasks across a range of open-domain environments.