Research

Sometimes I feel like a mathematical golden retriever. Tell me about your favorite research area, and I’ll find myself enjoying it too. But, in my field, computational uncertainty quantification, I’ve begun converging on an idea I’ve grown more and more excited about.

In particular, I want to understand how to design algorithms for modern inverse problems with highly informative, data-driven, priors. Although popular in its own right, I think the machine learning community has largely focused instead on learning the dynamics from data. That is, neural surrogates, reduced basis, etc. To be fair, this has proven wildly powerful.

But, physics is awesome¹! Especially, if it has structure to exploit, I can build extremely informed, fast, and understood solvers. Modern machine learning has proven undeniably powerful, but is it right to throw away understood dynamics? Especially when the prior seems a very natural object to inform with data (a.k.a. prior knowledge).

This diverges from the traditional selection of priors in classic inverse problems. For example, your Gaussian priors or Tikhonov regularization in the deterministic case. Unlike these, data-driven priors are extremely informative by design. They shrink something known as the effective dimension, or information dimension, of the inverse problem; the number of directions in which the data actually tells us something the prior didn’t already know. Some of our most scalable methods (Krylov solvers, randomized methods, etc.) pay only for this dimension.

But, distressingly, many of these methods function by exploiting the structure of a prior. Data-driven priors, are often black box and don’t satisfy the traditional convenient mathematical assumptions necessary to make many of these scalable methods efficient. I (often) cannot: decompose them, reason about their regularity, or sometimes even efficiently apply them. By the end of the next 2-3 years, I want to understand 1) how to build modern inverse problem solvers to handle these troublesome learned priors and 2) if I can design friendlier learned priors without sacrificing their expressibility. Currently, I’m in preparation of three research projects that attack this direction, with more coming. I hope to update this page by the end of this year with some progress!

Themes

Here’s a quick synopsis of some themes I’ve published in (with, of course, more in the pipeline):

Improved Algorithms for Machine Learning

I am interested in optimization and approximation algorithms for machine learning models. One recent avenue for this is looking for classical numerical analysis techniques that can clarify or improve modern training procedures. Recent work in this direction includes VPBoost (Chowdhary, Newman, et al. 2026), a gradient boosting method for separable smooth learners that combines variable projection with a trust-region view of second-order weak learning.

Scaling uncertainty quantification

A major part of my work concerns uncertainty quantification for Bayesian inverse problems, especially PDE-constrained problems with high- or infinite-dimensional unknowns. During my PhD, I worked on scalable ways to approximate, differentiate, and optimize information-based quantities such as information gain and expected information gain, with applications to sensitivity analysis and robust optimal experimental design. I published a few papers in this direction; see my work with Georg and coauthors (Chowdhary, Tong, et al. 2024), my work with Ahmed and Alen (Chowdhary, Attia, et al. 2026), and my thesis (Chowdhary 2025).

Mathematical software

I have a personal interest in mathematical research software. It’s key for our field to improve its transparency and reproducibility while also improving the speed of dissemination of information. In the past, I was a major maintainer for PyOED (Chowdhary, Ahmed, et al. 2024), which was a package for rapidly prototyping and benchmarking data assimilation and model-constrained optimal design methods.

Collaborators

One of my favorite parts of doing research is the many, incredible, scientists I’ve had the pleasure of working alongside. I’ve decided to keep a list of them, since each have made large impacts in how I think about mathematics.

Advisors

• Alen Alexanderian — North Carolina State University
• Ahmed Attia — Argonne National Laboratory
• Elizabeth Newman — Tufts University

Senior collaborators

• Eric Chi — University of Minnesota
• Julianne Chung — Emory University
• Arvind Saibaba — North Carolina State University
• Georg Stadler — New York University
• Shanyin Tong — University of Pennsylvania
• Deepanshu Verma — Clemson University
• Qiliang Wu — Ohio University

Postdoctoral collaborators

• Alberto Bucci — University of Edinburgh
• Federica Milinanni — Northwestern University
• Fan Tian — University of North Carolina at Chapel Hill

References

Chowdhary, Abhijit. 2025. “Scalable Uncertainty Quantification for Infinite-Dimensional Bayesian Inverse Problems.” {PhD} thesis.

Chowdhary, Abhijit, Shady E. Ahmed, and Ahmed Attia. 2024. “PyOED: An Extensible Suite for Data Assimilation and Model-Constrained Optimal Design of Experiments.” ACM Transactions on Mathematical Software 50 (2): 1–22. https://doi.org/10.1145/3653071.

Chowdhary, Abhijit, Ahmed Attia, and Alen Alexanderian. 2026. “Robust Optimal Experimental Design of Infinite-Dimensional Bayesian Nonlinear Inverse Problems.” SIAM Journal on Scientific Computing 48 (2): B262–88. https://doi.org/10.1137/24M1693921.

Chowdhary, Abhijit, Elizabeth Newman, and Deepanshu Verma. 2026. Boost Like a (Var)pro: Trust-Region Gradient Boosting via Variable Projection. https://arxiv.org/abs/2603.23658.

Chowdhary, Abhijit, Shanyin Tong, Georg Stadler, and Alen Alexanderian. 2024. “Sensitivity Analysis of the Information Gain in Infinite-Dimensional Bayesian Linear Inverse Problems.” International Journal for Uncertainty Quantification, ahead of print. https://doi.org/10.1615/Int.J.UncertaintyQuantification.2024051416.

Footnotes

1. Ah, well, sometimes it isn’t. In that case, yeah, let the GPUs rip.