James V. Roggeveen

I am an applied mathematician and engineer developing computational frameworks that connect physical modeling, experimental data, and optimization. My goal is to build robust, general-purpose tools that accelerate scientific discovery by closing the loop between theory, simulation, and experiment.

As a Postdoctoral Fellow at Harvard University with Prof. Michael Brenner, my research is focused on three main areas:

Meshless PDE Solvers: I have developed a new spectral method for solving PDE inverse problems on complex and irregular geometries. This differentiable, meshless solver avoids grid generation, demonstrates exponential convergence, and is a powerful engine for optimization problems, which I have applied to challenges like modeling Antarctic ice sheet viscosity from observational data.
Automated Model Discovery: I build tools that discover physical laws directly from data. I created a differentiable rheology framework that fits complex constitutive models to lab measurements and uses Bayesian methods to suggest the next best experiment to perform, accelerating materials characterization.
AI for Scientific Reasoning: To rigorously test the capabilities of modern AI, I co-led the development of the HARDMath2 benchmark. We pioneered a novel approach where graduate students in an applied math course created the problems, and I built the infrastructure to automatically parse and grade the AI-generated LaTeX solutions.

My Ph.D. in Mechanical Engineering at Princeton University, advised by Prof. Howard A. Stone, focused on particle dynamics in low-Reynolds-number flows and creating improved hydrodynamic models for characterizing the material properties of biological condensates using micropipette aspiration.

As a researcher and an engineer, my core competencies are in the development of data-driven mathematical models to predict, control, and optimize complex systems. My training has given me effective verbal, written, and visual communication skills, including conveying complex mathematical systems through animation. I am experienced with cross–discipline collaboration with diverse teams to implement engineering solutions in fast-paced environments and automation of data validation and analysis pipelines to improve efficiency in dealing with experimental data.

I am passionate about applying my skills in computational modeling, differentiable programming (JAX), and data-driven science to solve challenging problems in a collaborative environment.

news

Oct 29, 2025	Check out our preprint on solving inverse problems over PDEs with a meshelss basis fitting method, out now on arXiv.
Oct 28, 2025	Check out our preprint on fitting viscoelastic models to flow data using automatic differentiation, out now on arXiv.
Oct 06, 2025	Check out our preprint on CMT-Bench, an LLM benchmark focused on condensed matter physics built by a panel of experts, out now on arXiv.
Sep 18, 2025	Excited to announce that our paper HardMATH2 has been accepted at NeurIPS 2025!
Jul 28, 2025	Gave an invited seminar on “Beyond the Rheometer: Integrative Approaches to Modeling and Characterizing Complex Fluids” at Nanyang Technological University, Singapore.

selected publications

Learning constitutive models and rheology from partial flow measurements

Alp M. Sunol , James V. Roggeveen , Mohammed G. Alhashim , and 2 more authors

Oct 2025

Abs arXiv

Constitutive laws are at the core of fluid mechanics, relating the fluid stress to its deformation rate. Unlike Newtonian fluids, most industrial and biological fluids are non-Newtonian, exhibiting a nonlinear relation. Accurately characterizing this nonlinearity is essential for predicting flow behavior in real-world engineering and translational applications. Yet current methods fall short by relying on bulk rheometer data and simple fits that fail to capture behaviors relevant in complex geometries and flow conditions. Data-driven approaches can capture more complex behaviors, but lack interpretability or consistency. To close this gap, we leverage automatic differentiation to build an end-to-end framework for robust rheological learning. We develop a differentiable non-Newtonian fluid solver with a tensor basis neural network closure that learns stress directly from arbitrary flow measurements, such as velocimetry data. In parallel, we implement differentiable versions of major constitutive relations, enabling Bayesian model parametrization and selection from rheometer data. Our framework predicts flows in unseen geometries and ensures physical consistency and interpretability by matching neural network responses to known constitutive laws. Ultimately, this work lays the groundwork for advanced digital rheometry capable of comprehensively characterizing non-Newtonian and viscoelastic fluids under realistic in-situ or in-line operating conditions.
Meshless solutions of PDE inverse problems on irregular geometries

James V. Roggeveen , and Michael P. Brenner

Oct 2025

Abs arXiv

Solving inverse and optimization problems over solutions of nonlinear partial differential equations (PDEs) on complex spatial domains is a long-standing challenge. Here we introduce a method that parameterizes the solution using spectral bases on arbitrary spatiotemporal domains, whereby the basis is defined on a hyperrectangle containing the true domain. We find the coefficients of the basis expansion by solving an optimization problem whereby both the equations, the boundary conditions and any optimization targets are enforced by a loss function, building on a key idea from Physics-Informed Neural Networks (PINNs). Since the representation of the function natively has exponential convergence, so does the solution of the optimization problem, as long as it can be solved efficiently. We find empirically that the optimization protocols developed for machine learning find solutions with exponential convergence on a wide range of equations. The method naturally allows for the incorporation of data assimilation by including additional terms in the loss function, and for the efficient solution of optimization problems over the PDE solutions.
HARDMath2: A Benchmark for Applied Mathematics Built by Students as Part of a Graduate Class

James V. Roggeveen , Erik Y. Wang , Will Flintoft , and 42 more authors

May 2025

Abs arXiv Code

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 , 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.