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Generative AI has made remarkable progress in addressing various design challenges. One prominent area where generative AI could bring significant value is in engineering design. In particular, selecting an optimal set of components and their interfaces to create a mechanical system that meets design requirements is one of the most challenging and time-consuming tasks for engineers. This configuration design task is inherently challenging due to its categorical nature, multiple design requirements a solution must satisfy, and the reliance on physics simulations for evaluating potential solutions. These characteristics entail solving a combinatorial optimization problem with multiple constraints involving black-box functions. To address this challenge, we propose a deep generative model to predict the optimal combination of components and interfaces for a given design problem. To demonstrate our approach, we solve a gear train synthesis problem by first creating a synthetic dataset using a grammar, a parts catalogue, and a physics simulator. We then train a Transformer using this dataset, named GearFormer, which can not only generate quality solutions on its own, but also augment search methods such as an evolutionary algorithm and Monte Carlo tree search. We show that GearFormer outperforms such search methods on their own in terms of satisfying the specified design requirements with orders of magnitude faster generation time. Additionally, we showcase the benefit of hybrid methods that leverage both GearFormer and search methods, which further improve the quality of the solutions.

We introduce derangetropy, a novel functional measure designed to characterize the dynamics of information within probability distributions. Unlike scalar measures such as Shannon entropy, derangetropy offers a functional representation that captures the dispersion of information across the entire support of a distribution. By incorporating self-referential and periodic properties, it provides deeper insights into information dynamics governed by differential equations and equilibrium states. Through combinatorial justifications and empirical analysis, we demonstrate the utility of derangetropy in depicting distribution behavior and evolution, providing a new tool for analyzing complex and hierarchical systems in information theory.

This paper provides a comprehensive analysis of Generative AI (GenAI) potential to revolutionise software coding through increased efficiency and reduced time span for writing code. It acknowledges the transformative power of GenAI in software code generation, while also cautioning against the inherent risks of bias and errors if left unchecked. Emphasising the irreplaceable value of traditional programming, it posits that GenAI is not a replacement but a complementary tool for writing software code. Ethical considerations are paramount with the paper advocating for stringent ethical guidelines to ensure GenAI serves the greater good and does not compromise on accountability in writing software code. It suggests a balanced approach, combining human oversight with AI's capabilities, to mitigate risks and enhance reliability. The paper concludes by proposing guidelines for GenAI utilisation in coding, which will empower developers to navigate its complexities and employ it responsibly. This approach addresses current ethical concerns and sets a foundation for the judicious use of GenAI in the future, ensuring its benefits are harnessed effectively while maintaining moral integrity.

Detecting falls among the elderly and alerting their community responders can save countless lives. We design and develop a low-cost mobile robot that periodically searches the house for the person being monitored and sends an email to a set of designated responders if a fall is detected. In this project, we make three novel design decisions and contributions. First, our custom-designed low-cost robot has advanced features like omnidirectional wheels, the ability to run deep learning models, and autonomous wireless charging. Second, we improve the accuracy of fall detection for the YOLOv8-Pose-nano object detection network by 6% and YOLOv8-Pose-large by 12%. We do so by transforming the images captured from the robot viewpoint (camera height 0.15m from the ground) to a typical human viewpoint (1.5m above the ground) using a principally computed Homography matrix. This improves network accuracy because the training dataset MS-COCO on which YOLOv8-Pose is trained is captured from a human-height viewpoint. Lastly, we improve the robot controller by learning a model that predicts the robot velocity from the input signal to the motor controller., Comment: Accepted to IEEE AIM-2024

In this paper we extend some set theoretic concepts of numerical semigroups for arbitrary sub-semigroups of natural numbers. Then we characterized gapsets which leads to a more efficient computational approach towards numerical semigroups and finally we introduce the extension of gapsets and prove that the sequence of the number of gapsets of size $g$ is non-decreasing as a weak version of Bras-Amor\'os's conjecture., Comment: 14 pages, 2 algorithms

Uncertainty-aware controllers that guarantee safety are critical for safety critical applications. Among such controllers, Control Barrier Functions (CBFs) based approaches are popular because they are fast, yet safe. However, most such works depend on Gaussian Processes (GPs) or MC-Dropout for learning and uncertainty estimation, and both approaches come with drawbacks: GPs are non-parametric methods that are slow, while MC-Dropout does not capture aleatoric uncertainty. On the other hand, modern Bayesian learning algorithms have shown promise in uncertainty quantification. The application of modern Bayesian learning methods to CBF-based controllers has not yet been studied. We aim to fill this gap by surveying uncertainty quantification algorithms and evaluating them on CBF-based safe controllers. We find that model variance-based algorithms (for example, Deep ensembles, MC-dropout, etc.) and direct estimation-based algorithms (such as DEUP) have complementary strengths. Algorithms in the former category can only estimate uncertainty accurately out-of-domain, while those in the latter category can only do so in-domain. We combine the two approaches to obtain more accurate uncertainty estimates both in- and out-of-domain. As measured by the failure rate of a simulated robot, this results in a safer CBF-based robot controller.

In this work, we derive a sharp condition on the mass of the initial data for the global existence of the Chern-Simons-Schr\"odinger equation. As a corollary, we prove that if the strength of interaction is less than the Bogomolny bound, then, for a large enough mass of initial data, there exists a globally defined solution. On the other hand, for the interactions which are above the Bogomolny bound, the critical mass condition on the initial data for the global existence depends on the strength of the self-interacting field. Then, we show that the states with the initial critical mass and zero energy are standing wave solutions and globally well-posed. Moreover, they are static if the self-interacting field is large enough and non-static for small self-interacting field.

This work considers two related families of nonlinear and nonlocal problems in the plane $\mathbb{R}^2$. The first main result derives the general integrable solution to a generalized Liouville equation using the Wronskian of two coprime complex polynomials. The second main result concerns an application to a generalized Ladyzhenskaya-Gagliardo-Nirenberg interpolation inequality, with a single real parameter $\beta$ interpreted as the strength of a magnetic self-interaction. The optimal constant of the inequality and the corresponding minimizers of the quotient are studied and it is proved that for $\beta \ge 2$, for which the constant equals $2\pi\beta$, such minimizers only exist at quantized $\beta \in 2\mathbb{N}$ corresponding to nonlinear generalizations of Landau levels with densities solving the generalized Liouville equation. This latter problem originates from the study of self-dual vortex solitons in the abelian Chern-Simons-Higgs theory and from the average-field-Pauli effective theory of anyons, i.e. quantum particles with statistics intermediate to bosons and fermions. An immediate application is given to Keller-Lieb-Thirring stability bounds for a gas of such anyons which self-interact magnetically (vector nonlocal repulsion) as well as electrostatically (scalar local/point attraction), thus generalizing the stability theory of the 2D cubic nonlinear Schr\"odinger equation., Comment: 65 pages. V2: some minor corrections and added references

This research introduces DesignQA, a novel benchmark aimed at evaluating the proficiency of multimodal large language models (MLLMs) in comprehending and applying engineering requirements in technical documentation. Developed with a focus on real-world engineering challenges, DesignQA uniquely combines multimodal data-including textual design requirements, CAD images, and engineering drawings-derived from the Formula SAE student competition. Different from many existing MLLM benchmarks, DesignQA contains document-grounded visual questions where the input image and input document come from different sources. The benchmark features automatic evaluation metrics and is divided into segments-Rule Comprehension, Rule Compliance, and Rule Extraction-based on tasks that engineers perform when designing according to requirements. We evaluate state-of-the-art models (at the time of writing) like GPT-4o, GPT-4, Claude-Opus, Gemini-1.0, and LLaVA-1.5 against the benchmark, and our study uncovers the existing gaps in MLLMs' abilities to interpret complex engineering documentation. The MLLMs tested, while promising, struggle to reliably retrieve relevant rules from the Formula SAE documentation, face challenges in recognizing technical components in CAD images, and encounter difficulty in analyzing engineering drawings. These findings underscore the need for multimodal models that can better handle the multifaceted questions characteristic of design according to technical documentation. This benchmark sets a foundation for future advancements in AI-supported engineering design processes. DesignQA is publicly available at: https://github.com/anniedoris/design_qa/.

Requirements elicitation, a critical, yet time-consuming and challenging step in product development, often fails to capture the full spectrum of user needs. This may lead to products that fall short of expectations. This paper introduces a novel framework that leverages Large Language Models (LLMs) to automate and enhance the requirements elicitation process. LLMs are used to generate a vast array of simulated users (LLM agents), enabling the exploration of a much broader range of user needs and unforeseen use cases. These agents engage in product experience scenarios, through explaining their actions, observations, and challenges. Subsequent agent interviews and analysis uncover valuable user needs, including latent ones. We validate our framework with three experiments. First, we explore different methodologies for diverse agent generation, discussing their advantages and shortcomings. We measure the diversity of identified user needs and demonstrate that context-aware agent generation leads to greater diversity. Second, we show how our framework effectively mimics empathic lead user interviews, identifying a greater number of latent needs than conventional human interviews. Third, we showcase that LLMs can be used to analyze interviews, capture needs, and classify them as latent or not. Our work highlights the potential of using LLM agents to accelerate early-stage product development, reduce costs, and increase innovation.

The use of deep learning has become increasingly popular in reduced-order models (ROMs) to obtain low-dimensional representations of full-order models. Convolutional autoencoders (CAEs) are often used to this end as they are adept at handling data that are spatially distributed, including solutions to partial differential equations. When applied to unsteady physics problems, ROMs also require a model for time-series prediction of the low-dimensional latent variables. Long short-term memory (LSTM) networks, a type of recurrent neural network useful for modeling sequential data, are frequently employed in data-driven ROMs for autoregressive time-series prediction. When making predictions at unseen design points over long time horizons, error propagation is a frequently encountered issue, where errors made early on can compound over time and lead to large inaccuracies. In this work, we propose using bagging, a commonly used ensemble learning technique, to develop a fully data-driven ROM framework referred to as the CAE-eLSTM ROM that uses CAEs for spatial reconstruction of the full-order model and LSTM ensembles for time-series prediction. When applied to two unsteady fluid dynamics problems, our results show that the presented framework effectively reduces error propagation and leads to more accurate time-series prediction of latent variables at unseen points., Comment: Updated with revisions

The lattice Boltzmann method (LBM) has emerged as a prominent technique for solving fluid dynamics problems due to its algorithmic potential for computational scalability. We introduce XLB library, a Python-based differentiable LBM library based on the JAX platform. The architecture of XLB is predicated upon ensuring accessibility, extensibility, and computational performance, enabling scaling effectively across CPU, TPU, multi-GPU, and distributed multi-GPU or TPU systems. The library can be readily augmented with novel boundary conditions, collision models, or multi-physics simulation capabilities. XLB's differentiability and data structure is compatible with the extensive JAX-based machine learning ecosystem, enabling it to address physics-based machine learning, optimization, and inverse problems. XLB has been successfully scaled to handle simulations with billions of cells, achieving giga-scale lattice updates per second. XLB is released under the permissive Apache-2.0 license and is available on GitHub at https://github.com/Autodesk/XLB.

Motivated by recent interests in exploring excitonic condensate as the ground state of some narrow-bandgap semiconductors such as transition metal dichalcogenides and layered chalcogenide material Ta$_2$NiSe$_5$, in this work we theoretically study the dynamics of condensate in response to periodically driven laser fields with different polarizations and intensities. In particular, we consider laser light beams with bicircular and circular polarizations breaking the time-revesal symmetry, and linear polarization. We show that the amplitude of the condensate oscillates in time during irradiating by light with a magnitude depending on the light intensity. The dynamics survives even after the light is switched off. The phase mode however changes linearly with time for a condensate originating from purely electronic correlations. We further show that in the presence of electron-phonon coupling the linear-in-time behavior is replaced by a harmonically oscillating behavior, a manifestation of gapped phase modes due to relative band charge symmetry breaking. Furthermore, we show that the primarily electronic and primarily lattice cases corresponding to strong and weak electron-phonon coupling, respectively, reveal distinct dynamics of the condensate, an observation which can modify the optical response of an excitonic insulator by stimulating amplitude and phase modes in the former case., Comment: 12 pages, 6 figures

In this work, we study convection-diffusion equations in the cases of bounded drifts and drifts induced by the gradient of a potential. We define a new notion of solution and prove its existence and uniqueness. Furthermore, we show the conservation of mass, the convergence to the initial data, and the strong maximum principle.

Kitaev materials are a promising platform for the realization of quantum spin liquid states. The thermal Hall effect has emerged as a potential probe of exotic excitations within such states. In the Kitaev candidate material $\alpha$-RuCl$_{3}$, however, the thermal Hall conductivity $\kappa_{xy}$ has been attributed not only to exotic Majorana fermions or chiral magnons, but also to phonons. It has been shown theoretically that the former two types of heat carriers can generate a "planar" thermal Hall effect, whereby the magnetic field is parallel to the heat current, as observed experimentally, but it is unknown whether phonons also could. Here we show that a planar thermal Hall effect is present in another Kitaev candidate material, Na$_{2}$Co$_{2}$TeO$_{6}$. On the basis of a striking similarity between the temperature and field dependence of $\kappa_{xy}$ and that of the phonon-dominated thermal conductivity $\kappa_{xx}$, we argue that the planar thermal Hall effect in Na$_{2}$Co$_{2}$TeO$_{6}$ is generated by phonons. The phonon contributed planar $\kappa_{xy}$ also shows a strong sample dependence, which indicates an extrinsic origin of the mechanism. By conducting a complete study with different in-plane configurations of heat current $J$ and magnetic field $H$, $i.e.$ $H$ $\|$ $J$ and $H$ $\bot$ $J$, we observe a large difference in $\kappa_{xy}$ between these two configurations, which reveals that the direction of the heat current $J$ may play an important role in determining the planar thermal Hall effect. Our observation calls for a re-evaluation of the planar thermal Hall effect observed in $\alpha$-RuCl$_{3}$.