Omar Knio

Faris Hijazi, Somayah AlHarbi, Abdulaziz AlHussein et al.

The rapid advancements in Large Language Models (LLMs) have led to significant improvements in various natural language processing tasks. However, the evaluation of LLMs' legal knowledge, particularly in non-English languages such as Arabic, remains under-explored. To address this gap, we introduce ArabLegalEval, a multitask benchmark dataset for assessing the Arabic legal knowledge of LLMs. Inspired by the MMLU and LegalBench datasets, ArabLegalEval consists of multiple tasks sourced from Saudi legal documents and synthesized questions. In this work, we aim to analyze the capabilities required to solve legal problems in Arabic and benchmark the performance of state-of-the-art LLMs. We explore the impact of in-context learning and investigate various evaluation methods. Additionally, we explore workflows for generating questions with automatic validation to enhance the dataset's quality. We benchmark multilingual and Arabic-centric LLMs, such as GPT-4 and Jais, respectively. We also share our methodology for creating the dataset and validation, which can be generalized to other domains. We hope to accelerate AI research in the Arabic Legal domain by releasing the ArabLegalEval dataset and code: https://github.com/Thiqah/ArabLegalEval

K. Mustapha, K. Furati, O. M. Knio et al.

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

Loïc Giraldi, Olivier P. Le Maître, Ibrahim Hoteit et al.

Optimal dimensionality reduction methods are proposed for the Bayesian inference of a Gaussian linear model with additive noise in presence of overabundant data. Three different optimal projections of the observations are proposed based on information theory: the projection that minimizes the Kullback-Leibler divergence between the posterior distributions of the original and the projected models, the one that minimizes the expected Kullback-Leibler divergence between the same distributions, and the one that maximizes the mutual information between the parameter of interest and the projected observations. The first two optimization problems are formulated as the determination of an optimal subspace and therefore the solution is computed using Riemannian optimization algorithms on the Grassmann manifold. Regarding the maximization of the mutual information, it is shown that there exists an optimal subspace that minimizes the entropy of the posterior distribution of the reduced model; a basis of the subspace can be computed as the solution to a generalized eigenvalue problem; an a priori error estimate on the mutual information is available for this particular solution; and that the dimensionality of the subspace to exactly conserve the mutual information between the input and the output of the models is less than the number of parameters to be inferred. Numerical applications to linear and nonlinear models are used to assess the efficiency of the proposed approaches, and to highlight their advantages compared to standard approaches based on the principal component analysis of the observations.

Mohamad Abed El Rahman Hammoud, Naila Raboudi, Edriss S. Titi et al.

Data assimilation (DA) plays a pivotal role in diverse applications, ranging from climate predictions and weather forecasts to trajectory planning for autonomous vehicles. A prime example is the widely used ensemble Kalman filter (EnKF), which relies on linear updates to minimize variance among the ensemble of forecast states. Recent advancements have seen the emergence of deep learning approaches in this domain, primarily within a supervised learning framework. However, the adaptability of such models to untrained scenarios remains a challenge. In this study, we introduce a novel DA strategy that utilizes reinforcement learning (RL) to apply state corrections using full or partial observations of the state variables. Our investigation focuses on demonstrating this approach to the chaotic Lorenz '63 system, where the agent's objective is to minimize the root-mean-squared error between the observations and corresponding forecast states. Consequently, the agent develops a correction strategy, enhancing model forecasts based on available system state observations. Our strategy employs a stochastic action policy, enabling a Monte Carlo-based DA framework that relies on randomly sampling the policy to generate an ensemble of assimilated realizations. Results demonstrate that the developed RL algorithm performs favorably when compared to the EnKF. Additionally, we illustrate the agent's capability to assimilate non-Gaussian data, addressing a significant limitation of the EnKF.

Harethah Abu Shairah, Somayah AlHarbi, Abdulaziz AlHussein et al.

We introduce ALARB, a dataset and suite of tasks designed to evaluate the reasoning capabilities of large language models (LLMs) within the Arabic legal domain. While existing Arabic benchmarks cover some knowledge-intensive tasks such as retrieval and understanding, substantial datasets focusing specifically on multistep reasoning for Arabic LLMs, especially in open-ended contexts, are lacking. The dataset comprises over 13K commercial court cases from Saudi Arabia, with each case including the facts presented, the reasoning of the court, the verdict, as well as the cited clauses extracted from the regulatory documents. We define a set of challenging tasks leveraging this dataset and reflecting the complexity of real-world legal reasoning, including verdict prediction, completion of reasoning chains in multistep legal arguments, and identification of relevant regulations based on case facts. We benchmark a representative selection of current open and closed Arabic LLMs on these tasks and demonstrate the dataset's utility for instruction tuning. Notably, we show that instruction-tuning a modest 12B parameter model using ALARB significantly enhances its performance in verdict prediction and Arabic verdict generation, reaching a level comparable to that of GPT-4o.

Imran Pervez, Ricardo Pinto Lima, Omar Knio

We develop novel integrated learning and optimization (ILO) methodologies to solve economic dispatch (ED) and DC optimal power flow (DCOPF) problems for better economic operation. The optimization problem for ED is formulated with load being an unknown parameter while DCOPF consists of load and power transfer distribution factor (PTDF) matrix as unknown parameters. PTDF represents the incremental variations of real power on transmission lines which occur due to real power transfers between two regions. These values represent a linearized approximation of power flows over the transmission lines. We develop novel ILO formulations to solve post-hoc penalties in electricity market and line congestion problems using ED and DCOPF optimization formulations. Our proposed methodologies capture the real-time electricity market and line congestion behavior to train the regret function which eventually train unknown loads at different buses and line PTDF matrix to achieve the afore-mentioned post-hoc goals. The proposed methodology is compared to sequential learning and optimization (SLO) which train load and PTDF forecasts for accuracy rather than economic operation. Our experimentation prove the superiority of ILO in minimizing the post-hoc penalties in electricity markets and minimizing the line congestion thereby improving the economic operation with noticeable amount.

S. Allouch, M. Lucchesi, O. P. Le Maître et al.

This work explores different particle-based approaches to the simulation of one-dimensional fractional subdiffusion equations in unbounded domains. We rely on smooth particle approximations, and consider four methods for estimating the fractional diffusion term. The first method is based on direct differentiation of the particle representation, it follows the Riesz definition of the fractional derivative and results in a non-conservative scheme. The other three methods follow the particle strength exchange (PSE) methodology and are by construction conservative, in the sense that the total particle strength is time invariant. The first PSE algorithm is based on using direct differentiation to estimate the fractional diffusion flux, and exploiting the resulting estimates in an integral representation of the divergence operator. Meanwhile, the second one relies on the regularized Riesz representation of the fractional diffusion term to derive a suitable interaction formula acting directly on the particle representation of the diffusing field. A third PSE construction is considered that exploits the Green's function of the fractional diffusion equation. The performance of all four approaches is assessed for the case of a one-dimensional diffusion equation with constant diffusivity. This enables us to take advantage of known analytical solutions, and consequently conduct a detailed analysis of the performance of the methods. This includes a quantitative study of the various sources of error, namely filtering, quadrature, domain truncation, and time integration, as well as a space and time self-convergence analysis. These analyses are conducted for different values of the order of the fractional derivatives, and computational experiences are used to gain insight that can be used for generalization of the present constructions.