Supplement to Nan Chen's Army Research Office (ARO) research grant. Co-mentor with Nan Chen and Charlotte Moser (senior personnel member and research proposal co-author).

Program Information: The Army Educational Outreach Program (AEOP) Internship is organised and sponsored by the Army Research Office (ARO) of the U.S. Army Combat Capabilities Development Command Army Research Laboratory (DEVCOM ARL). This project is conducted under AEOP's Undergraduate Research Apprenticeship Program (URAP) and High School Apprenticeship Program (HSAP) to support the summer research for one undergraduate student and one high school student in 2026.

Interns: URAP — Daria Sevriukova, University of Wisconsin–Madison & HSAP — Bryant Yu, Madison West High School

Project Description: These AEOP projects focus on two distinct subtopics.

• Temporal Causal Networks in Nonlinear Dynamical Systems via Assimilative Causal Inference (URAP, undergraduate project). We develop a dynamical-systems framework for temporal causal networks using assimilative causal inference. We assemble instantaneous causal contributions into temporally evolving weighted directed graphs, enabling the identification of regime-dependent causal fingerprints, causal reorganizations during transitions, and low-dimensional coherent structures in causal space. Initial studies use the Lorenz '63 system to establish the geometry of temporal causality, with outcomes guiding the investigation of a topographic flow model to reveal causal pathways associated with zonal–blocked regime transitions, including blocking onset and decay. This project is still ongoing independently.

  • Final Project Report

  • Reading Material

• Exploring the Causality of El Niño–Southern Oscillation (ENSO) Complexity Using Conceptual Models (HSAP, high school project).  We analyse the many facets of ENSO complexity and diversity using low-dimensional conceptual models which capture large-scale statistical and dynamical features critical for applied sciences. We conduct systematic comparisons between multiple stochastic conceptual models with the help of real observational data.

  • Final Project Report

  • Reading Material

Co-mentor with Pouria Behnoudfar (project mentor).

Program Information: The MXM Lab was founded in 2021, and aims to enhance and support undergraduate research within the Department of Mathematics and the University of Wisconsin.

Undergraduate Team Members: Tan Bui, Cody McKenna, Singer Xing, Daniel Youngberg.

Project Description: Dynamical systems model many real-world phenomena, such as weather conditions and the nervous system. However, many operational models that emulate these systems are caught up in complexity and fail to model basic statistical quantities such as regional average temperatures, seasonal variabilities, energy spectra, or correlations between activating neural tissue and their recovering inhibitor. We develop a latent data assimilation methodology that integrates statistical measurements with forecasts to model complex dynamics while keeping predictions physically grounded. In this framework, we train an autoencoder that maps a high-dimensional physical state into a lower-dimensional latent space, and augment the latent state with coarse statistical observations about the system before refining the predictions with an ensemble Kalman filter. We apply this framework to two separate models. First, the stochastic Lorenz ’96 model, a geophysical model of atmospheric energy, where we achieve accurate latent dynamics. Second, the stochastically coupled FitzHugh–Nagumo model, where we obtain accurate activation–recovery behavior by using spike and fast–slow correlation statistics as the observed statistical measurements.

Final Project Report

Reading Material: 

• Project outline

• Bridging idealized and operational models: an explainable AI framework for Earth system emulators

• The Kalman Filter and the Ensemble Kalman Filter

• A Brief Note on the Stochastic Lorenz ’96 and Coupled FitzHugh–Nagumo Models and the Euler–Maruyama Method