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The aim of this manuscript is to address non-linear differential equations of the Lane Emden equation of second order using the shifted Legendre neural network (SLNN) method. Here all the equations are classified as singular initial value problems. To manage the singularity challenge, we employ an artificial neural network method. The approach utilizes a neural network of a single layer, where the hidden layer is omitted by enlarge the input using shifted Legendre polynomials. We apply a feed forward neural network model along with the principle of error back propagation. The effectiveness of the Legendre Neural Network model is demonstrated through LaneEmden equations.

The study of low regularity Cauchy data for nonlinear dispersive PDEs has successfully been achieved using modulation spaces $M^{p,q}$ in recent years. In this paper, we study the inhomogeneous nonlinear Schr\"odinger equation (INLS) $$iu_t + \Delta u\pm |x|^{-b}|u|^{\alpha}u=0,$$ where $\alpha, b>0,$ on whole space $\mathbb R^n$ in modulation spaces. In the subcritical regime $(0<\alpha< \frac{4-2b}{n}),$ we establish local well-posedness in $L^{2}+M^{\alpha+2,\frac{\alpha+2}{\alpha+1}}( \supset L^2 + H^s \ \text{for} \ s>\frac{n\alpha}{2(\alpha+2)}).$ By adapting Bourgain's high-low decomposition method, we establish global well-posedness in $M^{p,\frac{p}{p-1}}$ with $2

In this paper, we explore the feasibility of leveraging large language models (LLMs) to automate or otherwise assist human raters with identifying harmful content including hate speech, harassment, violent extremism, and election misinformation. Using a dataset of 50,000 comments, we demonstrate that LLMs can achieve 90% accuracy when compared to human verdicts. We explore how to best leverage these capabilities, proposing five design patterns that integrate LLMs with human rating, such as pre-filtering non-violative content, detecting potential errors in human rating, or surfacing critical context to support human rating. We outline how to support all of these design patterns using a single, optimized prompt. Beyond these synthetic experiments, we share how piloting our proposed techniques in a real-world review queue yielded a 41.5% improvement in optimizing available human rater capacity, and a 9--11% increase (absolute) in precision and recall for detecting violative content.

Visual program synthesis is a promising approach to exploit the reasoning abilities of large language models for compositional computer vision tasks. Previous work has used few-shot prompting with frozen LLMs to synthesize visual programs. Training an LLM to write better visual programs is an attractive prospect, but it is unclear how to accomplish this. No dataset of visual programs for training exists, and acquisition of a visual program dataset cannot be easily crowdsourced due to the need for expert annotators. To get around the lack of direct supervision, we explore improving the program synthesis abilities of an LLM using feedback from interactive experience. We propose a method where we exploit existing annotations for a vision-language task to improvise a coarse reward signal for that task, treat the LLM as a policy, and apply reinforced self-training to improve the visual program synthesis ability of the LLM for that task. We describe a series of experiments on object detection, compositional visual question answering, and image-text retrieval, and show that in each case, the self-trained LLM outperforms or performs on par with few-shot frozen LLMs that are an order of magnitude larger. Website: https://zaidkhan.me/ViReP, Comment: CVPR 2024

Although planning is a crucial component of the autonomous driving stack, researchers have yet to develop robust planning algorithms that are capable of safely handling the diverse range of possible driving scenarios. Learning-based planners suffer from overfitting and poor long-tail performance. On the other hand, rule-based planners generalize well, but might fail to handle scenarios that require complex driving maneuvers. To address these limitations, we investigate the possibility of leveraging the common-sense reasoning capabilities of Large Language Models (LLMs) such as GPT4 and Llama2 to generate plans for self-driving vehicles. In particular, we develop a novel hybrid planner that leverages a conventional rule-based planner in conjunction with an LLM-based planner. Guided by commonsense reasoning abilities of LLMs, our approach navigates complex scenarios which existing planners struggle with, produces well-reasoned outputs while also remaining grounded through working alongside the rule-based approach. Through extensive evaluation on the nuPlan benchmark, we achieve state-of-the-art performance, outperforming all existing pure learning- and rule-based methods across most metrics. Our code will be available at https://llmassist.github.io., Comment: 15 pages, 8 figures, 7 tables

The recent progress in language-based open-vocabulary object detection can be largely attributed to finding better ways of leveraging large-scale data with free-form text annotations. Training such models with a discriminative objective function has proven successful, but requires good positive and negative samples. However, the free-form nature and the open vocabulary of object descriptions make the space of negatives extremely large. Prior works randomly sample negatives or use rule-based techniques to build them. In contrast, we propose to leverage the vast knowledge built into modern generative models to automatically build negatives that are more relevant to the original data. Specifically, we use large-language-models to generate negative text descriptions, and text-to-image diffusion models to also generate corresponding negative images. Our experimental analysis confirms the relevance of the generated negative data, and its use in language-based detectors improves performance on two complex benchmarks. Code is available at \url{https://github.com/xiaofeng94/Gen-Enhanced-Negs}., Comment: Accepted to CVPR 2024. The supplementary document included