The weak form is a ubiquitous, well-studied, and widely-utilized mathematical tool in modern computational and applied mathematics. In this work we provide a survey of both the history and recent developments for several fields in which the weak form can play a critical role. In particular, we highlight several recent advances in weak form versions of equation learning, parameter estimation, and coarse graining, which offer surprising noise robustness, accuracy, and computational efficiency. We note that this manuscript is a companion piece to our October 2024 SIAM News article of the same name. Here we provide more detailed explanations of mathematical developments as well as a more complete list of references. Lastly, we note that the software with which to reproduce the results in this manuscript is also available on our group's GitHub website https://github.com/MathBioCU .

Though current long-context large language models (LLMs) have demonstrated impressive capacities in answering user questions based on extensive text, the lack of citations in their responses makes user verification difficult, leading to concerns about their trustworthiness due to their potential hallucinations. In this work, we aim to enable long-context LLMs to generate responses with fine-grained sentence-level citations, improving their faithfulness and verifiability. We first introduce LongBench-Cite, an automated benchmark for assessing current LLMs' performance in Long-Context Question Answering with Citations (LQAC), revealing considerable room for improvement. To this end, we propose CoF (Coarse to Fine), a novel pipeline that utilizes off-the-shelf LLMs to automatically generate long-context QA instances with precise sentence-level citations, and leverage this pipeline to construct LongCite-45k, a large-scale SFT dataset for LQAC. Finally, we train LongCite-8B and LongCite-9B using the LongCite-45k dataset, successfully enabling their generation of accurate responses and fine-grained sentence-level citations in a single output. The evaluation results on LongBench-Cite show that our trained models achieve state-of-the-art citation quality, surpassing advanced proprietary models including GPT-4o.

Robotic grippers are the end effector in the robot system of handling any task which used for performing various operations for the purpose of industrial application and hazardous tasks.In this paper, we developed the mathematical model for multi fingers robotics grippers. we are concerned with Jamia'shand which is developed in Robotics Lab, Mechanical Engineering Deptt, Faculty of Engg & Technolgy, Jamia Millia Islamia, India. This is a tendon-driven gripper each finger having three DOF having a total of 11 DOF. The term tendon is widely used to imply belts, cables, or similar types of applications. It is made up of three fingers and a thumb. Every finger and thumb has one degree of freedom. The power transmission mechanism is a rope and pulley system. Both hands have similar structures. Aluminum from the 5083 families was used to make this product. The gripping force can be adjusted we have done the kinematics, force, and dynamic analysis by developing a Mathematical model for the four-finger robotics grippers and their thumb. we focused it control motions in X and Y Displacements with the angular positions movements and we make the force analysis of the four fingers and thumb calculate the maximum weight, force, and torque required to move it with mass. Draw the force -displacements graph which shows the linear behavior up to 250 N and shows nonlinear behavior beyond this. and required Dmin of wire is 0.86 mm for grasping the maximum 1 kg load also developed the dynamic model (using energy )approach lagrangian method to find it torque required to move the fingers.

The nonlinearities of soft robots bring control challenges like hysteresis but also provide them with computational capacities. This paper introduces a fuzzy pneumatic physical reservoir computing (FPRC) model for feedforward hysteresis compensation in motion tracking control of soft actuators. Our method utilizes a pneumatic bending actuator as a physical reservoir with nonlinear computing capacities to control another pneumatic bending actuator. The FPRC model employs a Takagi-Sugeno (T-S) fuzzy model to process outputs from the physical reservoir. In comparative evaluations, the FPRC model shows equivalent training performance to an Echo State Network (ESN) model, whereas it exhibits better test accuracies with significantly reduced execution time. Experiments validate the proposed FPRC model's effectiveness in controlling the bending motion of the pneumatic soft actuator with open and closed-loop control systems. The proposed FPRC model's robustness against environmental disturbances has also been experimentally verified. To the authors' knowledge, this is the first implementation of a physical system in the feedforward hysteresis compensation model for controlling soft actuators. This study is expected to advance physical reservoir computing in nonlinear control applications and extend the feedforward hysteresis compensation methods for controlling soft actuators.

The time-consuming nature of training and deploying complicated Machine and Deep Learning (DL) models for a variety of applications continues to pose significant challenges in the field of Machine Learning (ML). These challenges are particularly pronounced in the federated domain, where optimizing models for individual nodes poses significant difficulty. Many methods have been developed to tackle this problem, aiming to reduce training expenses and time while maintaining efficient optimisation. Three suggested strategies to tackle this challenge include Active Learning, Knowledge Distillation, and Local Memorization. These methods enable the adoption of smaller models that require fewer computational resources and allow for model personalization with local insights, thereby improving the effectiveness of current models. The present study delves into the fundamental principles of these three approaches and proposes an advanced Federated Learning System that utilises different Personalisation methods towards improving the accuracy of AI models and enhancing user experience in real-time NG-IoT applications, investigating the efficacy of these techniques in the local and federated domain. The results of the original and optimised models are then compared in both local and federated contexts using a comparison analysis. The post-analysis shows encouraging outcomes when it comes to optimising and personalising the models with the suggested techniques.

The rapid advancement in large language models (LLMs) comes with a significant increase in their parameter size, presenting challenges for adaptation and fine-tuning. Parameter-efficient fine-tuning (PEFT) methods are widely used to adapt LLMs for downstream tasks efficiently. In this paper, we propose Singular Values and Orthonormal Regularized Singular Vectors Adaptation, or SORSA, a novel PEFT method. We introduce a method to analyze the variation of the parameters by performing singular value decomposition (SVD) and discuss and analyze SORSA's superiority in minimizing the alteration in the SVD aspect. Each SORSA adapter consists of two main parts: trainable principal singular weights $W_p = U_p \Sigma_p V^\top_p$, and frozen residual weights $W_r = U_r \Sigma_r V^\top_r$. These parts are initialized by performing SVD on pre-trained weights. Moreover, we implement and analyze an orthonormal regularizer, which could effectively transfer the scaling information into $\Sigma_p$ and ultimately allows the training process to be more efficient. SORSA adapters could be merged during inference, thus eliminating any inference latency. After all, SORSA shows a faster convergence than PiSSA and LoRA in our experiments. On the MATH benchmark, Llama 2 7B adapted using SORSA achieved 10.36% accuracy, outperforming LoRA (5.50%), Full FT (7.22%), and PiSSA (7.44%). On the GSM-8K benchmark, SORSA achieved 56.03% accuracy, surpassing LoRA (42.30%), Full FT (49.05%), and PiSSA (53.07%). We conclude that SORSA offers a new perspective on parameter-efficient fine-tuning, demonstrating remarkable performance. The code is available at https://github.com/Gunale0926/SORSA.

Multiobjective optimization problems (MOPs) are prevalent in machine learning, with applications in multi-task learning, learning under fairness or robustness constraints, etc. Instead of reducing multiple objective functions into a scalar objective, MOPs aim to optimize for the so-called Pareto optimality or Pareto set learning, which involves optimizing more than one objective function simultaneously, over models with millions of parameters. Existing benchmark libraries for MOPs mainly focus on evolutionary algorithms, most of which are zeroth-order methods that do not effectively utilize higher-order information from objectives and cannot scale to large-scale models with millions of parameters. In light of the above gap, this paper introduces LibMOON, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair benchmark, and is open-sourced for the community.

This letter presents a novel multi-robot task allocation and path planning method that considers robots' maximum range constraints in large-sized workspaces, enabling robots to complete the assigned tasks within their range limits. Firstly, we developed a fast path planner to solve global paths efficiently. Subsequently, we propose an innovative auction-based approach that integrates our path planner into the auction phase for reward computation while considering the robots' range limits. This method accounts for extra obstacle-avoiding travel distances rather than ideal straight-line distances, resolving the coupling between task allocation and path planning. Additionally, to avoid redundant computations during iterations, we implemented a lazy auction strategy to speed up the convergence of the task allocation. Finally, we validated the proposed method's effectiveness and application potential through extensive simulation and real-world experiments. The implementation code for our method will be available at https://github.com/wuuya1/RangeTAP.

As autonomous vehicles become more prevalent, highly accurate and efficient systems are increasingly critical to improve safety, performance, and energy consumption. Efficient management of energy-reliability tradeoffs in these systems demands the ability to predict various conditions during vehicle operations. With the promising improvement of Large Language Models (LLMs) and the emergence of well-known models like ChatGPT, unique opportunities for autonomous vehicle-related predictions have been provided in recent years. This paper proposed MAPS using LLMs as map reader co-drivers to predict the vital parameters to set during the autonomous vehicle operation to balance the energy-reliability tradeoff. The MAPS method demonstrates a 20% improvement in navigation accuracy compared to the best baseline method. MAPS also shows 11% energy savings in computational units and up to 54% in both mechanical and computational units.

Computing atomic-scale properties of chemically disordered materials requires an efficient exploration of their vast configuration space. Traditional approaches such as Monte Carlo or Special Quasirandom Structures either entail sampling an excessive amount of configurations or do not ensure that the configuration space has been properly covered. In this work, we propose a novel approach where generative machine learning is used to yield a representative set of configurations for accurate property evaluation and provide accurate estimations of atomic-scale properties with minimal computational cost. Our method employs a specific type of variational autoencoder with inverse roles for the encoder and decoder, enabling the application of an unsupervised active learning scheme that does not require any initial training database. The model iteratively generates configuration batches, whose properties are computed with conventional atomic-scale methods. These results are then fed back into the model to estimate the partition function, repeating the process until convergence. We illustrate our approach by computing point-defect formation energies and concentrations in (U, Pu)O2 mixed-oxide fuels. In addition, the ML model provides valuable insights into the physical factors influencing the target property. Our method is generally applicable to explore other properties, such as atomic-scale diffusion coefficients, in ideally or non-ideally disordered materials like high-entropy alloys.