Ben Brown

Geoff Vasil, Daniel Lecoanet, Keaton Burns et al.

This paper presents a method for the accurate and efficient computations on scalar, vector and tensor fields in three-dimensional spherical polar coordinates. The methods uses spin-weighted spherical harmonics in the angular directions and rescaled Jacobi polynomials in the radial direction. For the 2-sphere, spin-weighted harmonics allow for automating calculations in a fashion as similar to Fourier series as possible. Derivative operators act as wavenumber multiplication on a set of spectral coefficients. After transforming the angular directions, a set of orthogonal tensor rotations put the radially dependent spectral coefficients into individual spaces each obeying a particular regularity condition at the origin. These regularity spaces have remarkably simple properties under standard vector-calculus operations, such as \textit{grad} and \textit{div}. We use a hierarchy of rescaled Jacobi polynomials for a basis on these regularity spaces. It is possible to select the Jacobi-polynomial parameters such that all relevant operators act in a minimally banded way. Altogether, the geometric structure allows for the accurate and efficient solution of general partial differential equations in the unit ball.

Irina Jurenka, Markus Kunesch, Kevin R. McKee et al.

A major challenge facing the world is the provision of equitable and universal access to quality education. Recent advances in generative AI (gen AI) have created excitement about the potential of new technologies to offer a personal tutor for every learner and a teaching assistant for every teacher. The full extent of this dream, however, has not yet materialised. We argue that this is primarily due to the difficulties with verbalising pedagogical intuitions into gen AI prompts and the lack of good evaluation practices, reinforced by the challenges in defining excellent pedagogy. Here we present our work collaborating with learners and educators to translate high level principles from learning science into a pragmatic set of seven diverse educational benchmarks, spanning quantitative, qualitative, automatic and human evaluations; and to develop a new set of fine-tuning datasets to improve the pedagogical capabilities of Gemini, introducing LearnLM-Tutor. Our evaluations show that LearnLM-Tutor is consistently preferred over a prompt tuned Gemini by educators and learners on a number of pedagogical dimensions. We hope that this work can serve as a first step towards developing a comprehensive educational evaluation framework, and that this can enable rapid progress within the AI and EdTech communities towards maximising the positive impact of gen AI in education.

Gemini Team, Rohan Anil, Sebastian Borgeaud et al.

This report introduces a new family of multimodal models, Gemini, that exhibit remarkable capabilities across image, audio, video, and text understanding. The Gemini family consists of Ultra, Pro, and Nano sizes, suitable for applications ranging from complex reasoning tasks to on-device memory-constrained use-cases. Evaluation on a broad range of benchmarks shows that our most-capable Gemini Ultra model advances the state of the art in 30 of 32 of these benchmarks - notably being the first model to achieve human-expert performance on the well-studied exam benchmark MMLU, and improving the state of the art in every one of the 20 multimodal benchmarks we examined. We believe that the new capabilities of the Gemini family in cross-modal reasoning and language understanding will enable a wide variety of use cases. We discuss our approach toward post-training and deploying Gemini models responsibly to users through services including Gemini, Gemini Advanced, Google AI Studio, and Cloud Vertex AI.