Research

My research involves the design and analysis of efficient machine learning algorithms that are tailor-made for scientific and other types of continuum data. This work is motivated by scientific computing tasks that involve complex physical systems or inverse problems, where the data is often heterogeneous, noisy, incomplete, and limited in number. I deploy the methodologies arising from my research in several application areas, including medical imaging, weather forecasting, Earth systems modeling, and materials science. My work enables scientists and engineers to solve previously intractable high-dimensional problems, increase the level of trust placed in machine-learned predictions, and achieve target accuracy thresholds with fewer training data and lower cost.

Some research topics that I have been thinking about recently include: operator learning for parametrized partial differential equations, statistical and stochastic inverse problems, non-Euclidean data analysis, generative modeling of probability distributions, approximate Bayesian inference and uncertainty quantification, optimal sampling and experimental design, and data assimilation for dynamical systems.

Scientific Machine Learning

Scientific machine learning (SciML) blends modern ideas from artificial intelligence with more traditional scientific computing paradigms. It promises to accelerate simulation, discovery, and decision-making in the physical sciences. A popular line of work in SciML is operator learning, which is a data-driven framework for learning mappings between function spaces that remain robust under discretization refinement. This approach enables the construction of fast surrogates for expensive forward simulations that typically involve partial differential equations (PDEs). However, the mathematical theory of operator learning lags behind its empirical success. Can we identify an underlying PDE by only inputting forcing terms and observing corresponding solutions? What probability distribution should training data be sampled from to best approximate a target operator? How can we improve sample complexity by exploiting domain knowledge?

To answer these questions, my work develops mathematical and statistical foundations of SciML that inform practical strategies for building reliable and data-efficient models which are also robust to distribution shifts.

Research in SciML often requires functional analysis, statistical learning theory, high performance computing, and approximation theory.

In operator learning, model inputs and outputs are functions instead of finite vectors

Inverse Problems

Inverse problems concern the recovery of hidden parameters from indirect and noisy measurements. A primary challenge is that small perturbations to the measurements can cause large reconstruction errors. Certain imaging problems exhibit severe forms of this instability, such as electrical impedance tomography. Operator learning algorithms hold potential to solve nonlinear inverse problems more accurately than existing inversion methods. However, these operator learning algorithms have primarily been tailored to well-posed forward problems. In contrast, inverse problems necessarily require regularization to combat inherent ill-posedness. There is no consensus yet about how to address inverse problem instability through training or architecture design.

My research program aims to unveil fundamental tradeoffs among different data-driven inversion strategies and establish quantitative error bounds in terms of noise level, number of measurements, and number of training data. It also develops new ways to design and process complicated measurement data modalities.

Inverse problems research typically involves optimization, partial differential equations, numerical analysis, and dynamical systems.

Uncertainty Quantification

Uncertainty quantification (UQ) focuses on the systematic characterization, propagation, and reduction of uncertainties in complex computational and statistical models to enable reliable predictions and decisions. A major pillar of UQ is Bayesian inference, which turns prior knowledge and noisy observations into calibrated uncertainty: a posterior distribution over unknown parameters. Bayesian inference has close connections to inverse problems and is increasingly made more computationally tractable with SciML techniques. Conversely, UQ can also benefit SciML, especially in safety-critical decision-making settings where models must report not only point predictions but also trustworthy confidence measures.

My work on UQ aims to more accurately probe the prior-to-posterior map and accelerate the solution of Bayesian inverse problems. This is especially relevant for sequential optimal experimental design or Bayesian data assimilation applications, e.g., probabilistic weather forecasting.

UQ research often involves high-dimensional probability, sampling and measure transport, stochastic processes, and Bayesian statistics.

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