Biological, ecological, and atmospheric science research typically relies on differential equation models that are based on theory developed in those respective fields, which are then calibrated with data. While there are established methods for fitting these models to data, it is much trickier to quantify uncertainty in these models. For example, the data could be noisy, be from different sources, or some variables may be unaccounted for.
My research has two primary motivations. First, bringing statistical rigor and notions of uncertainty to these applications is crucial to have a deeper understanding of what the data says. If scientists and researchers better understand the uncertainty of the model and data, then decisions will be better informed, results from scientific experiments can be more clearly identified, and future models can aim to minimize uncertainty going forward. On the other hand, evaluating these differential equation models are often computationally expensive. Additionally, many procedures, such as optimization or system monitoring, requires regular evaluation of these expensive operations. Therefore, a second motivation of my research is developing scalable methods to performance this inference, allowing scientists to perform these expensive operations much more frequently and quickly.
My research encompasses a few areas:
Simulation-Based Inference in Differential Equation Models
Differential equation models are extensively used in life sciences to make predictions on our health or environment under various conditions. Bayesian inference calibrates the parameters of these models with data to ensure they are realistic and produces accurate measurements of uncertainty. However, complex differential equation models are computationally infeasible for most traditional Bayesian approaches. Simulation-based inference is the subfield of Bayesian computation that relies on repeatedly simulating data from different differential equation models to identify which models and parameter values are realistic.Amortized Inference
In some situations, it is crucial to evaluate the differential equation models many times, such as in optimization or when systems must be regularly monitored. Often differential equation models are computationally expensive, taking hours to days for each evaluation, which is only exacerbated when this needs to be repeated. Amortized inference is when a framework for the differential equation, such as the differential equation itself, or a surrogate, is executed many times, and the Bayesian inference is learned using data-driven models. Then, this inference can be repeated with those computationally cheaper learned mappings.Physics-Informed Learning with Heterogeneous Data Sources
Physics-informed learning is when physics is used to improve the predictive power of data-driven models by leveraging information from the Biological, Physical, Chemical, or Ecological differential equation models. Usually, this is done by adding a regularization term that ensures solutions look more similar to solutions to differential equations, but my projects are more focused on heterogeneous data sources, where a data-driven model is trained with two different datasets—one of them from a differential equation model and another consists of observed data. Bayesian-inspired approaches are used to train a data-driven model on observations while being partially informed by those differential equation models.Scalable High-Dimensional Inference
Because my work incorporates Bayesian methodologies with complex, mathematical models and data from different sources, the problems are often high-dimensional and computationally expensive. Therefore, I tend to also be interested in research involving dimension reduction and computational speedups.Statistical Shape Analysis
Statistical shape analysis is an approach that uses statistical models to analyze the shape of a sample of objects. The shape of objects is represented as a collection of coordinates in 2D or 3D space after standardizing for size, orientation, and shift. Often these statistical shape models project every object to a lower-dimensional subspace before performing inference, such as prediction or effect estimation of a treatment.Although I haven't formally started research in these areas, I'm keen to start new projects involving these subjects: