#01Jul 22, 2026
cs.LG
PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs
Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee and 3 more
Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability. Kolmogorov Arnold Networks (KANs) mitigate these limitations because their learnable spline activations are structurally aligned with the piecewise-polynomial bases of classical discretizations. However, the way a PDE is cast into a loss functional is as decisive as the choice of approximator: strong-form residual minimization requires high-order derivatives and heavily weighted losses, the energy (Bubnov-Galerkin) form is restricted to self-adjoint operators and, as we show, collapses to a trivial solution for parameter-identification problems, and boundary integral forms require a known fundamental solution. We propose PG-KINN, a physics-informed KAN built on a Petrov-Galerkin formulation in which the trial space is a KAN and the test space is an independent, compactly supported, piecewise-polynomial space evaluated with Gauss-Legendre quadrature. Integration by parts lowers the differentiation order while retaining applicability to general non-self-adjoint, nonlinear, and inverse problems; the localized test functions turn the global residual into a set of element-wise weak residuals with favorable conditioning. On a suite of benchmarks spanning crack singularities, stress concentration, Neo-Hookean hyperelasticity, inverse parameter identification in heterogeneous media, and complex geometries, PG-KINN consistently outperforms legacy MLP baselines and state-of-the-art KAN-based strong/energy/inverse formulations (PIKAN). These results position the Petrov-Galerkin coupling of KAN trial spaces and polynomial test spaces as a robust and accurate route for AI-based computational mechanics.
#02Jul 22, 2026
cs.CL
LKValues: Aligning Large Language Models with Sri Lankan Societal Values
Nethmi Muthugala, Supryadi, Surangika Ranathunga and 7 more
Value alignment of Large Language Models (LLMs) has been shown to be culturally biased toward Western norms. This results in the mishandling of local values in multilingual societies such as Sri Lanka that have their unique cultural dynamics. Existing benchmarks overlook Sri Lankan-contextualized values in its official language Sinhala, hindering culturally sensitive evaluation and fine-tuning. To bridge this gap, we propose LKValues, the first survey-grounded resource suite for Sri Lankan value alignment. From a trilingual survey of 205 respondents, blending adapted global frameworks and LLM-elicited local constructs, we derive 40 majority-endorsed societal values. Using these values, we construct LKvaluesIT, a Sinhala-English news-derived instruction corpus containing 150k scenario-based instances, and LKvaluesBench, a value-sensitive evaluation benchmark of 1,000 instances. We evaluate a set of proprietary and open-weight LLMs with LKvaluesBench. We fine-tune three open-weight base models (Qwen3.5-4B-Base, Qwen3.5-9B-Base, and Aya-Expanse-8B-Base). Our experiments show that newer and larger LLMs still exhibit low-resource and cultural value-alignment gaps. LKValues fine-tuning improves Qwen-family models in English and Sinhala, reducing invalid outputs and cross-lingual disparities, though gains remain model-family dependent. These highlight LKValues efficacy in embedding Sri Lankan values, offering a replicable pipeline for low-resource, country-specific pluralist value alignment. The dataset is publicly available at https://github.com/NextME14/LKValues.
#03Jul 22, 2026
stat.ML
Adaptive deep nonparametric regression from dependent data under covariate shift
William Kengne, Ehud Mossa Ockegna
Covariate shift often occurs because, in many real applications, the source and the target observations may be generated from different distributions. In this case, the standard metric under the source distribution is not appropriate. This paper considers deep neural network estimators for nonparametric quantile and Huber regression under covariate shift and from dependent observations. We deal with a generalized Bernstein-type inequality that is satisfied by many classical models, including i.i.d. observations, $φ$-mixing, strong mixing, and $\mathcal{C}$-mixing processes. To perform the covariate shift phenomenon, we propose a sparse-penalized deep neural network (SPDNN) estimator that takes into account the discrepancy between the source and target distributions of the data. When the density ratio (between the source and target distributions of the covariate) is unknown, a two steps pre-training procedure is carried out: the first step is devoted to the construction of a least squares SPDNN estimator of the density ratio; which is used in the second step to perform a pre-training reweighted SPDNN estimator of the regression function. For both the quantile and the Huber regression, non-asymptotic error bounds of the proposed SPDNN estimators are established in the class of Hölder smooth functions. These estimators can adaptively attain (up to a logarithmic factor) the minimax optimal convergence rate from i.i.d. data as well as from several classical time series models.
#04Jul 22, 2026
cs.CL
Notes to Self: Can LLMs Benefit from Experiential Abstractions?
Chang Liu, Xinyu Li, Artur Dubrawski
Humans distill experience into reusable abstractions, e.g., strategies and cautionary reminders, and apply them to gradually solve problems more effectively. We study whether Large Language Models (LLMs) can similarly benefit from such experiential abstractions. From LLMs' solution traces on the MATH training set, a stronger teacher or the LLMs themselves extract natural-language abstractions into a retrievable library. We explore two usage modes: (1) inference-time retrieval and (2) reinforcement learning (RL) with abstraction-augmented training prompts. Experiential abstractions improve LLM performance on mathematical and logical reasoning benchmarks. Self-extracted abstractions match teacher-extracted ones, and our abstraction usage framework can transfer to other datasets and models. These findings suggest LLMs can extract and apply experiential abstractions much as humans leverage distilled experience.
#05Jul 22, 2026
math.OC
Decentralized Online Riemannian Optimization for Strongly Geodesically Convex Functions
Zhanyuan Cai, Emre Sahinoglu, Shahin Shahrampour
We study decentralized online optimization for strongly geodesically convex (strongly g-convex) losses on Riemannian manifolds with bounded sectional curvature, including positively curved manifolds. In centralized Riemannian optimization, strong g-convexity tightens the optimal regret from $O(\sqrt{T})$ to $O(\log T)$, where $T$ is the time horizon; in the decentralized Riemannian setting, however, existing methods address only g-convex losses, leaving the strongly g-convex regime unexplored. One challenge is that the required decaying step size in the centralized regime is incompatible with existing network-error analyses, which typically assume a fixed step size. First, we provide a general network-error analysis for time-varying schedules. Next, we build on this analysis to establish the first $O(\log T)$ static regret bound for decentralized online Riemannian gradient descent, matching the minimax-optimal rate for strongly-convex Euclidean online optimization. Finally, we prove the same $O(\log T)$ regret bound for the two-point bandit feedback setting using novel strong subconvexity arguments for the smoothed versions of the loss functions.