CDS&E: Data-Driven Nonlinear Model Reduction for High-dimensional Multi-scale Stochastic Dynamical Systems

Mathematical models of complex stochastic systems arise across the sciences, including physical chemistry, biology, physics, and atmospheric science. Important questions about such systems, including which long-lived states they settle into and how often they transition between states, are hard to answer by direct simulation because the systems span many time scales and capturing a single rare event can require billions of tiny numerical steps. This project will develop new mathematical and computational tools that learn compact, accurate models of long-term behavior from short bursts of simulation data, allowing researchers to identify stable states, estimate transition rates, and predict long-time behavior far more efficiently than is currently possible. These tools will accelerate molecular and chemical simulation, biological modeling, and other data-intensive areas of computational science. The work will also support training of graduate and undergraduate students, a new statistical machine learning course at the University at Albany, open-source software, public educational materials in topological data analysis and data science, and Saturday outreach programs that bring interactive artificial intelligence activities to high school students from under-resourced communities. The project will develop data-driven nonlinear model reduction methods for high-dimensional, multiscale stochastic dynamical systems whose effective long-time behavior is concentrated on unknown low-dimensional invariant manifolds. The methods will use ensembles of short, parallelizable trajectories from a black-box simulator to estimate the local geometry, drift, and diffusion of the slow dynamics. The research pursues three interconnected goals. First, the investigators will design adaptive exploration algorithms that combine a tangent-space biased explorer with topological data analysis to escape metastable basins and detect uncovered regions of the slow manifold. Second, they will construct local simulators that learn effective stochastic dynamics in overlapping coordinate charts using chart autoencoders, with dynamics-adapted regularization that keeps learned tangent spaces, curvature, drift, and diffusion consistent across charts, enabling intrinsic time stepping that avoids repeated reprojection in the ambient space. Third, they will establish theoretical guarantees, including generalization error and sample complexity bounds with rates depending only on the intrinsic dimension of the manifold. The methods will be validated on synthetic benchmarks and on biomolecular systems including alanine dipeptide, larger peptides, and coarse-grained polymers. Anticipated contributions include faster exploration of rare transitions, sharper long-time statistics, and reusable open-source software that bridges topological and geometric machine learning with stochastic dynamics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.