HARDMath: A Benchmark Dataset for Challenging Problems in Applied Mathematics

Advanced applied mathematics problems are underrepresented in existing Large Language Model (LLM) benchmark datasets. To address this, we introduce HARDMath, a dataset inspired by a graduate course on asymptotic methods, featuring challenging applied mathematics problems that require analytical approximation techniques. These problems demand a combination of mathematical reasoning, computational tools, and subjective judgment, making them difficult for LLMs. Our framework auto-generates a large number of problems with solutions validated against numerical ground truths. We evaluate both open- and closed-source LLMs on HARDMath-mini, a sub-sampled test set of 366 problems, as well as on 40 word problems formulated in applied science contexts. Even leading closed-source models like GPT-4 achieve only 43.8% overall accuracy with few-shot Chain-of-Thought prompting, and all models demonstrate significantly lower performance compared to results on existing mathematics benchmark datasets. We additionally conduct a detailed error analysis to gain insights into the failure cases of LLMs. These results demonstrate limitations of current LLM performance on advanced graduate-level applied math problems and underscore the importance of datasets like HARDMath to advance mathematical abilities of LLMs.

Engineering morphogenesis of cell clusters with differentiable programming

Understanding the rules underlying organismal development is a major unsolved problem in biology. Each cell in a developing organism responds to signals in its local environment by dividing, excreting, consuming, or reorganizing, yet how these individual actions coordinate over a macroscopic number of cells to grow complex structures with exquisite functionality is unknown. Here we use recent advances in automatic differentiation to discover local interaction rules and genetic networks that yield emergent, systems-level characteristics in a model of development. We consider a growing tissue with cellular interactions are mediated by morphogen diffusion, differential cell adhesion and mechanical stress. Each cell has an internal genetic network that it uses to make decisions based on its local environment. We show that one can simultaneously learn parameters governing the cell interactions and the genetic network for complex developmental scenarios, including the symmetry breaking of an embryo from an initial cell, the creation of emergent chemical gradients,homogenization of growth via mechanical stress, programmed growth into a prespecified shape, and the ability to repair from damage. When combined with recent experimental advances measuring spatio-temporal dynamics and gene expression of cells in a growing tissue, the methodology outlined here offers a promising path to unravelling the cellular basis of development.

Quantum Many-Body Physics Calculations with Large Language Models

Large language models (LLMs) have demonstrated an unprecedented ability to perform complex tasks in multiple domains, including mathematical and scientific reasoning. We demonstrate that with carefully designed prompts, LLMs can accurately carry out key calculations in research papers in theoretical physics. We focus on a broadly used approximation method in quantum physics: the Hartree-Fock method, requiring an analytic multi-step calculation deriving approximate Hamiltonian and corresponding self-consistency equations. To carry out the calculations using LLMs, we design multi-step prompt templates that break down the analytic calculation into standardized steps with placeholders for problem-specific information. We evaluate GPT-4's performance in executing the calculation for 15 research papers from the past decade, demonstrating that, with correction of intermediate steps, it can correctly derive the final Hartree-Fock Hamiltonian in 13 cases and makes minor errors in 2 cases. Aggregating across all research papers, we find an average score of 87.5 (out of 100) on the execution of individual calculation steps. Overall, the requisite skill for doing these calculations is at the graduate level in quantum condensed matter theory. We further use LLMs to mitigate the two primary bottlenecks in this evaluation process: (i) extracting information from papers to fill in templates and (ii) automatic scoring of the calculation steps, demonstrating good results in both cases. The strong performance is the first step for developing algorithms that automatically explore theoretical hypotheses at an unprecedented scale.

Using Large Language Models to Accelerate Communication for Users with Severe Motor Impairments

Finding ways to accelerate text input for individuals with profound motor impairments has been a long-standing area of research. Closing the speed gap for augmentative and alternative communication (AAC) devices such as eye-tracking keyboards is important for improving the quality of life for such individuals. Recent advances in neural networks of natural language pose new opportunities for re-thinking strategies and user interfaces for enhanced text-entry for AAC users. In this paper, we present SpeakFaster, consisting of large language models (LLMs) and a co-designed user interface for text entry in a highly-abbreviated form, allowing saving 57% more motor actions than traditional predictive keyboards in offline simulation. A pilot study with 19 non-AAC participants typing on a mobile device by hand demonstrated gains in motor savings in line with the offline simulation, while introducing relatively small effects on overall typing speed. Lab and field testing on two eye-gaze typing users with amyotrophic lateral sclerosis (ALS) demonstrated text-entry rates 29-60% faster than traditional baselines, due to significant saving of expensive keystrokes achieved through phrase and word predictions from context-aware LLMs. These findings provide a strong foundation for further exploration of substantially-accelerated text communication for motor-impaired users and demonstrate a direction for applying LLMs to text-based user interfaces.

Neural General Circulation Models

General circulation models (GCMs) are the foundation of weather and climate prediction. GCMs are physics-based simulators which combine a numerical solver for large-scale dynamics with tuned representations for small-scale processes such as cloud formation. Recently, machine learning (ML) models trained on reanalysis data achieved comparable or better skill than GCMs for deterministic weather forecasting. However, these models have not demonstrated improved ensemble forecasts, or shown sufficient stability for long-term weather and climate simulations. Here we present the first GCM that combines a differentiable solver for atmospheric dynamics with ML components, and show that it can generate forecasts of deterministic weather, ensemble weather and climate on par with the best ML and physics-based methods. NeuralGCM is competitive with ML models for 1-10 day forecasts, and with the European Centre for Medium-Range Weather Forecasts ensemble prediction for 1-15 day forecasts. With prescribed sea surface temperature, NeuralGCM can accurately track climate metrics such as global mean temperature for multiple decades, and climate forecasts with 140 km resolution exhibit emergent phenomena such as realistic frequency and trajectories of tropical cyclones. For both weather and climate, our approach offers orders of magnitude computational savings over conventional GCMs. Our results show that end-to-end deep learning is compatible with tasks performed by conventional GCMs, and can enhance the large-scale physical simulations that are essential for understanding and predicting the Earth system.

Speech Intelligibility Classifiers from 550k Disordered Speech Samples

We developed dysarthric speech intelligibility classifiers on 551,176 disordered speech samples contributed by a diverse set of 468 speakers, with a range of self-reported speaking disorders and rated for their overall intelligibility on a five-point scale. We trained three models following different deep learning approaches and evaluated them on ~94K utterances from 100 speakers. We further found the models to generalize well (without further training) on the TORGO database (100% accuracy), UASpeech (0.93 correlation), ALS-TDI PMP (0.81 AUC) datasets as well as on a dataset of realistic unprompted speech we gathered (106 dysarthric and 76 control speakers,~2300 samples).

Learning to correct spectral methods for simulating turbulent flows

Despite their ubiquity throughout science and engineering, only a handful of partial differential equations (PDEs) have analytical, or closed-form solutions. This motivates a vast amount of classical work on numerical simulation of PDEs and more recently, a whirlwind of research into data-driven techniques leveraging machine learning (ML). A recent line of work indicates that a hybrid of classical numerical techniques with machine learning can offer significant improvements over either approach alone. In this work, we show that the choice of the numerical scheme is crucial when incorporating physics-based priors. We build upon Fourier-based spectral methods, which are considerably more efficient than other numerical schemes for simulating PDEs with smooth and periodic solutions. Specifically, we develop ML-augmented spectral solvers for three model PDEs of fluid dynamics, which improve upon the accuracy of standard spectral solvers at the same resolution. We also demonstrate a handful of key design principles for combining machine learning and numerical methods for solving PDEs.

Context-Aware Abbreviation Expansion Using Large Language Models

Motivated by the need for accelerating text entry in augmentative and alternative communication (AAC) for people with severe motor impairments, we propose a paradigm in which phrases are abbreviated aggressively as primarily word-initial letters. Our approach is to expand the abbreviations into full-phrase options by leveraging conversation context with the power of pretrained large language models (LLMs). Through zero-shot, few-shot, and fine-tuning experiments on four public conversation datasets, we show that for replies to the initial turn of a dialog, an LLM with 64B parameters is able to exactly expand over 70% of phrases with abbreviation length up to 10, leading to an effective keystroke saving rate of up to about 77% on these exact expansions. Including a small amount of context in the form of a single conversation turn more than doubles abbreviation expansion accuracies compared to having no context, an effect that is more pronounced for longer phrases. Additionally, the robustness of models against typo noise can be enhanced through fine-tuning on noisy data.

Using a Cross-Task Grid of Linear Probes to Interpret CNN Model Predictions On Retinal Images

We analyze a dataset of retinal images using linear probes: linear regression models trained on some "target" task, using embeddings from a deep convolutional (CNN) model trained on some "source" task as input. We use this method across all possible pairings of 93 tasks in the UK Biobank dataset of retinal images, leading to ~164k different models. We analyze the performance of these linear probes by source and target task and by layer depth. We observe that representations from the middle layers of the network are more generalizable. We find that some target tasks are easily predicted irrespective of the source task, and that some other target tasks are more accurately predicted from correlated source tasks than from embeddings trained on the same task.

Comparing Supervised Models And Learned Speech Representations For Classifying Intelligibility Of Disordered Speech On Selected Phrases

Automatic classification of disordered speech can provide an objective tool for identifying the presence and severity of speech impairment. Classification approaches can also help identify hard-to-recognize speech samples to teach ASR systems about the variable manifestations of impaired speech. Here, we develop and compare different deep learning techniques to classify the intelligibility of disordered speech on selected phrases. We collected samples from a diverse set of 661 speakers with a variety of self-reported disorders speaking 29 words or phrases, which were rated by speech-language pathologists for their overall intelligibility using a five-point Likert scale. We then evaluated classifiers developed using 3 approaches: (1) a convolutional neural network (CNN) trained for the task, (2) classifiers trained on non-semantic speech representations from CNNs that used an unsupervised objective [1], and (3) classifiers trained on the acoustic (encoder) embeddings from an ASR system trained on typical speech [2]. We found that the ASR encoder's embeddings considerably outperform the other two on detecting and classifying disordered speech. Further analysis shows that the ASR embeddings cluster speech by the spoken phrase, while the non-semantic embeddings cluster speech by speaker. Also, longer phrases are more indicative of intelligibility deficits than single words.