arXiv subjects

Michael P. Brenner

Publications and source records attributed to Michael P. Brenner.

At least 37 records · Page 2Linked to original sources

arXiv2025

EEFSUVA: A New Mathematical Olympiad Benchmark

Recent breakthroughs have spurred claims that large language models (LLMs) match gold medal Olympiad to graduate level proficiency on mathematics benchmarks. In this work, we examine these claims in detail and assess the extent to which current benchmarks capture genuine LLM mathematical reasoning. The composition of these benchmarks, primarily drawing from the International Mathematics Olympiad (IMO) and related competitions, may overstate models reasoning ability due to potential data contamination and a narrow focus on familiar problem types. To enable a more holistic assessment of mathematical understanding, we introduce EEFSUVA, a novel benchmark curated from under circulated regional and national Olympiads of Eastern Europe and the countries from the former Soviet Union. These contests feature problems of comparable difficulty to the IMO and are renowned for demanding nonstandard problem-solving techniques, yet their problems are far less prevalent in online corpora. Preliminary results suggest that even state-of-the-art LLMs exhibit a notable performance decline on EEFSUVA relative to other Olympiad-style benchmarks. These findings also suggest the potential importance of broader evaluation datasets for a fuller assessment of mathematical reasoning and for guiding future model development.

cs.CL↗

arXiv2025

HARDMath2: A Benchmark for Applied Mathematics Built by Students as Part of a Graduate Class

Large language models (LLMs) have shown remarkable progress in mathematical problem-solving, but evaluation has largely focused on problems that have exact analytical solutions or involve formal proofs, often overlooking approximation-based problems ubiquitous in applied science and engineering. To fill this gap, we build on prior work and present HARDMath2, a dataset of 211 original problems covering the core topics in an introductory graduate applied math class, including boundary-layer analysis, WKB methods, asymptotic solutions of nonlinear partial differential equations, and the asymptotics of oscillatory integrals. This dataset was designed and verified by the students and instructors of a core graduate applied mathematics course at Harvard. We build the dataset through a novel collaborative environment that challenges students to write and refine difficult problems consistent with the class syllabus, peer-validate solutions, test different models, and automatically check LLM-generated solutions against their own answers and numerical ground truths. Evaluation results show that leading frontier models still struggle with many of the problems in the dataset, highlighting a gap in the mathematical reasoning skills of current LLMs. Importantly, students identified strategies to create increasingly difficult problems by interacting with the models and exploiting common failure modes. This back-and-forth with the models not only resulted in a richer and more challenging benchmark but also led to qualitative improvements in the students' understanding of the course material, which is increasingly important as we enter an age where state-of-the-art language models can solve many challenging problems across a wide domain of fields.

cs.LG↗

arXiv2025

Fitting Coarse-Grained Models to Macroscopic Experimental Data via Automatic Differentiation

Developing physics-based models for molecular simulation requires fitting many unknown parameters to diverse experimental datasets. Traditionally, this process is piecemeal and difficult to reproduce, leading to a fragmented landscape of models. Here, we establish a systematic, extensible framework for fitting coarse-grained molecular models to macroscopic experimental data by leveraging recently developed methods for computing low-variance gradient estimates with automatic differentiation. Using a widely validated DNA force field as an exemplar, we develop methods for optimizing structural, mechanical, and thermodynamic properties across a range of simulation techniques, including enhanced sampling and external forcing, spanning micro- and millisecond timescales. We highlight how gradients enable efficient sensitivity analyses that yield physical insight. We then demonstrate the broad applicability of these techniques by optimizing diverse biomolecular systems, including RNA and DNA-protein hybrid models. We show how conflict-free gradient methods from multi-task learning can be adapted to impose multiple constraints simultaneously without compromising accuracy. This approach provides a foundation for transparent, reproducible, community-driven force field development, accelerating progress in molecular modeling.

physics.bio-ph↗

arXiv2025

FEABench: Evaluating Language Models on Multiphysics Reasoning Ability

Building precise simulations of the real world and invoking numerical solvers to answer quantitative problems is an essential requirement in engineering and science. We present FEABench, a benchmark to evaluate the ability of large language models (LLMs) and LLM agents to simulate and solve physics, mathematics and engineering problems using finite element analysis (FEA). We introduce a comprehensive evaluation scheme to investigate the ability of LLMs to solve these problems end-to-end by reasoning over natural language problem descriptions and operating COMSOL Multiphysics$^\circledR$, an FEA software, to compute the answers. We additionally design a language model agent equipped with the ability to interact with the software through its Application Programming Interface (API), examine its outputs and use tools to improve its solutions over multiple iterations. Our best performing strategy generates executable API calls 88% of the time. LLMs that can successfully interact with and operate FEA software to solve problems such as those in our benchmark would push the frontiers of automation in engineering. Acquiring this capability would augment LLMs' reasoning skills with the precision of numerical solvers and advance the development of autonomous systems that can tackle complex problems in the real world. The code is available at https://github.com/google/feabench

cs.AI↗

arXiv2025

Towards AI-assisted Academic Writing

We present components of an AI-assisted academic writing system including citation recommendation and introduction writing. The system recommends citations by considering the user's current document context to provide relevant suggestions. It generates introductions in a structured fashion, situating the contributions of the research relative to prior work. We demonstrate the effectiveness of the components through quantitative evaluations. Finally, the paper presents qualitative research exploring how researchers incorporate citations into their writing workflows. Our findings indicate that there is demand for precise AI-assisted writing systems and simple, effective methods for meeting those needs.

cs.AI↗

arXiv2025

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 mediated by morphogen diffusion, cell adhesion and mechanical stress. Each cell has an internal genetic network that is used to make decisions based on the cell's local environment. We show that one can learn the parameters governing cell interactions in the form of interpretable genetic networks for complex developmental scenarios, including directed axial elongation, cell type homeostasis via chemical signaling and homogenization of growth via mechanical stress. 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 unraveling the cellular bases of development.

q-bio.CB↗

arXiv2024

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.

cs.LG↗

arXiv2024

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.

physics.comp-ph↗

arXiv2024

Inferring interaction potentials from stochastic particle trajectories

Accurate interaction potentials between microscopic components such as colloidal particles or cells are crucial to understanding a range of processes, including colloidal crystallization, bacterial colony formation, and cancer metastasis. Even in systems where the precise interaction mechanisms are unknown, effective interactions can be measured to inform simulation and design. However, these measurements are difficult and time-intensive, and often require conditions that are drastically different from in situ conditions of the system of interest. Moreover, existing methods of measuring interparticle potentials rely on constraining a small number of particles at equilibrium, placing limits on which interactions can be measured. We introduce a method for inferring interaction potentials directly from trajectory data of interacting particles. We explicitly solve the equations of motion to find a form of the potential that maximizes the probability of observing a known trajectory. Our method is valid for systems both in and out of equilibrium, is well-suited to large numbers of particles interacting in typical system conditions, and does not assume a functional form of the interaction potential. We apply our method to infer the interactions of colloidal spheres from experimental data, successfully extracting the range and strength of a depletion interaction from the motion of the particles.

cond-mat.soft↗

arXiv2024

Neural General Circulation Models for Weather and Climate

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.

physics.ao-ph↗

arXiv2023

Proofreading mechanism for colloidal self-assembly

Designing components that can robustly self-assemble into structures with biological complexity is a grand challenge for material science. Proofreading and error correction is required to improve assembly yield beyond equilibrium limits, using energy to avoid kinetic traps in the energy landscape. Here we introduce an explicit two staged proofreading scheme for patchy particle colloidal assemblies that substantially improves assembly yield and robustness. The first stage implements local rules whereby particles increase their binding strengths when they detect a local environment corresponding to a desired target. The second stage corrects remaining errors, adding a reverse pathway inspired by kinetic proofreading. The scheme shows significant yield improvements, eliminating kinetic traps, giving a much broader temperature range with high yield. Additionally, the scheme is robust against quenched disorder in the components. Our findings illuminate a pathway for advancing programmable design of synthetic living materials, potentially fostering the synthesis of novel biological materials and functional behaviors.

cond-mat.soft↗

arXiv2023

Programmable patchy particles for materials design

Direct design of complex functional materials would revolutionize technologies ranging from printable organs to novel clean energy devices. However, even incremental steps towards designing functional materials have proven challenging. If the material is constructed from highly complex components, the design space of materials properties rapidly becomes too computationally expensive to search. On the other hand, very simple components such as uniform spherical particles are not powerful enough to capture rich functional behavior. Here, we introduce a differentiable materials design model with components that are simple enough to design yet powerful enough to capture complex materials properties: rigid bodies composed of spherical particles with directional interactions (patchy particles). We showcase the method with self-assembly designs ranging from open lattices to self-limiting clusters, all of which are notoriously challenging design goals to achieve using purely isotropic particles. By directly optimizing over the location and interaction of the patches on patchy particles using gradient descent, we dramatically reduce the computation time for finding the optimal building blocks.

cond-mat.soft↗

arXiv2023

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.

cs.HC↗

arXiv2023

Recurrent flow patterns as a basis for turbulence: predicting statistics from structures

A dynamical systems approach to turbulence envisions the flow as a trajectory through a high-dimensional state space transiently visiting the neighbourhoods of unstable simple invariant solutions (E. Hopf, Commun. Appl. Maths 1, 303, 1948). The hope has always been to turn this appealing picture into a predictive framework where the statistics of the flow follows from a weighted sum of the statistics of each simple invariant solution. Two outstanding obstacles have prevented this goal from being achieved: (1) paucity of known solutions and (2) the lack of a rational theory for predicting the required weights. Here we describe a method to substantially solve these problems, and thereby provide the first compelling evidence that the PDFs of a fully developed turbulent flow can be reconstructed with a set of unstable periodic orbits. Our new method for finding solutions uses automatic differentiation, with high-quality guesses constructed by minimising a trajectory-dependent loss function. We use this approach to find hundreds of new solutions in turbulent, two-dimensional Kolmogorov flow. Robust statistical predictions are then computed by learning weights after converting a turbulent trajectory into a Markov chain for which the states are individual solutions, and the nearest solution to a given snapshot is determined using a deep convolutional autoencoder. To our knowledge, this is the first time the PDFs of a spatio-temporally-chaotic system have been successfully reproduced with a set of simple invariant states, and provides a fascinating connection between self-sustaining dynamical processes and the more well-known statistical properties of turbulence.

physics.flu-dyn↗

arXiv2023

Exact coherent structures in two-dimensional turbulence identified with convolutional autoencoders

Convolutional autoencoders are used to deconstruct the changing dynamics of two-dimensional Kolmogorov flow as $Re$ is increased from weakly chaotic flow at $Re=40$ to a chaotic state dominated by a domain-filling vortex pair at $Re=400$. The highly accurate embeddings allow us to visualise the evolving structure of state space and are interpretable using `latent Fourier analysis' (Page {\em et. al.}, \emph{Phys. Rev. Fluids} \textbf{6}, 2021). Individual latent Fourier modes decode into vortical structures with a streamwise lengthscale controlled by the latent wavenumber, $l$, with only a small number $l \lesssim 8$ required to accurately represent the flow. Latent Fourier projections reveal a detached class of bursting events at $Re=40$ which merge with the low-dissipation dynamics as $Re$ is increased to $100$. We use doubly- ($l=2$) or triply- ($l=3$) periodic latent Fourier modes to generate guesses for UPOs (unstable periodic orbits) associated with high-dissipation events. While the doubly-periodic UPOs are representative of the high-dissipation dynamics at $Re=40$, the same class of UPOs move away from the attractor at $Re=100$ -- where the associated bursting events typically involve larger-scale ($l=1$) structure too. At $Re=400$ an entirely different embedding structure is formed within the network in which no distinct representations of small-scale vortices are observed; instead the network embeds all snapshots based around a large-scale template for the condensate. We use latent Fourier projections to find an associated `large-scale' UPO which we believe to be a finite-$Re$ continuation of a solution to the Euler equations.

physics.flu-dyn↗

arXiv2023

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 and 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 known to be more efficient than other numerical schemes for simulating PDEs with smooth and periodic solutions. Specifically, we develop ML-augmented spectral solvers for three common PDEs of fluid dynamics. Our models are more accurate (2-4x) than standard spectral solvers at the same resolution but have longer overall runtimes (~2x), due to the additional runtime cost of the neural network component. We also demonstrate a handful of key design principles for combining machine learning and numerical methods for solving PDEs.

cs.LG↗

arXiv2023

Energy transfer and vortex structures: Visualizing the incompressible turbulent energy cascade

The transfer of kinetic energy from large to small scales is a hallmark of turbulent flows. Yet, a precise mechanistic description of this transfer, which is expected to occur via an energy cascade, is still missing. Several conceptually simple configurations with vortex tubes have been proposed as a testing ground to understand the energy cascade. Here, we focus on incompressible flows and compare the energy transfer occurring in a statistically steady homogeneous isotropic turbulent (HIT) flow with the generation of fine-scale motions in configurations involving vortex tubes. We start by filtering the velocity field in bands of wavenumbers distributed logarithmically, which allows us to study energy transfer in Fourier space and also visualize the energy cascade in real space. In the case of a statistically steady HIT flow at a moderate Reynolds number, our numerical results do not reveal any significant correlation between regions of intense energy transfers and vorticity or strain, filtered in corresponding wavenumber bands, nor any simple self-similar process. In comparison, in the transient turbulent flow obtained from the interaction between two antiparallel vortex tubes, we observe a qualitatively simpler organization of the intense structures, as well as of the energy transfer. The process leading to vortex reconnection via flattening of interacting vortex cores in two intense ribbons of vorticity, also appears as qualitatively different from the physics of HIT. Our results indicate that the specific properties of the transient flows affect the way energy is transferred, and may not be representative of HIT.

physics.flu-dyn↗

arXiv2023

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).

eess.AS↗