A Careful Examination of Large Language Model Performance on Grade School Arithmetic

Large language models (LLMs) have achieved impressive success on many benchmarks for mathematical reasoning. However, there is growing concern that some of this performance actually reflects dataset contamination, where data closely resembling benchmark questions leaks into the training data, instead of true reasoning ability. To investigate this claim rigorously, we commission Grade School Math 1000 (GSM1k). GSM1k is designed to mirror the style and complexity of the established GSM8k benchmark, the gold standard for measuring elementary mathematical reasoning. We ensure that the two benchmarks are comparable across important metrics such as human solve rates, number of steps in solution, answer magnitude, and more. When evaluating leading open- and closed-source LLMs on GSM1k, we observe accuracy drops of up to 13%, with several families of models (e.g., Phi and Mistral) showing evidence of systematic overfitting across almost all model sizes. At the same time, many models, especially those on the frontier, (e.g., Gemini/GPT/Claude) show minimal signs of overfitting. Further analysis suggests a positive relationship (Spearman's r^2=0.32) between a model's probability of generating an example from GSM8k and its performance gap between GSM8k and GSM1k, suggesting that many models may have partially memorized GSM8k.

Parametric Matrix Models

We present a general class of machine learning algorithms called parametric matrix models. Parametric matrix models are based on matrix equations, and the design is motivated by the efficiency of reduced basis methods for approximating solutions of parametric equations. The dependent variables can be defined implicitly or explicitly, and the equations may use algebraic, differential, or integral relations. Parametric matrix models can be trained with empirical data only, and no high-fidelity model calculations are needed. While originally designed for scientific computing, parametric matrix models are universal function approximators that can be applied to general machine learning problems. After introducing the underlying theory, we apply parametric matrix models to a series of different challenges that show their performance for a wide range of problems. For all the challenges tested here, parametric matrix models produce accurate results within a computational framework that allows for parameter extrapolation and interpretability.

Artificial Intelligence and Machine Learning in Nuclear Physics

Advances in artificial intelligence/machine learning methods provide tools that have broad applicability in scientific research. These techniques are being applied across the diversity of nuclear physics research topics, leading to advances that will facilitate scientific discoveries and societal applications. This Review gives a snapshot of nuclear physics research which has been transformed by artificial intelligence and machine learning techniques.

Self-learning Emulators and Eigenvector Continuation

Emulators that can bypass computationally expensive scientific calculations with high accuracy and speed can enable new studies of fundamental science as well as more potential applications. In this work we focus on solving a system of constraint equations efficiently using a new machine learning approach that we call self-learning emulation. A self-learning emulator is an active learning protocol that can rapidly solve a system of equations over some range of control parameters. The key ingredient is a fast estimate of the emulator error that becomes progressively more accurate as the emulator improves. This acceleration is possible because the emulator itself is used to estimate the error, and we illustrate with two examples. The first uses cubic spline interpolation to find the roots of a polynomial with variable coefficients. The second example uses eigenvector continuation to find the eigenvectors and eigenvalues of a large Hamiltonian matrix that depends on several control parameters. We envision future applications of self-learning emulators for solving systems of algebraic equations, linear and nonlinear differential equations, and linear and nonlinear eigenvalue problems.