Projects

Title: Learning Dynamics on Invariant Measures Using PDE-Constrained Optimization

Collaborators: Robert S. Martin (DEVCOM Army Research Laboratory) and Yunan Yang (Cornell University)

Abstract: We extend the methodology in [Yang et al., 2023] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.

Title: An Unstructured Mesh Approach to Nonlinear Noise Reduction for Coupled Systems

Collaborators: Aaron Kirtland (Brown University), Marianne DeBritto (University of Michigan), Megan Osborne (Rensselaer Polytechnic Institute), Casey Johnson (University of California, Los Angelos), Robert S. Martin (DEVCOM Army Research Laboratory), Samuel J. Araki (Jacobs Technology Inc., AFRL), Daniel Q. Eckhardt (In-Space Propulsion Branch, AFRL)

Abstract: To address noise inherent in electronic data acquisition systems and real-world sources, [Araki et al., 2021] demonstrated a grid-based nonlinear technique to remove noise from a chaotic signal, leveraging a clean high-fidelity signal from the same dynamical system and ensemble averaging in multidimensional phase space. This method achieved denoising of a time series data with 100% added noise but suffered in regions of low data density. To improve this grid-based method, here an unstructured mesh based on triangulations and Voronoi diagrams is used to accomplish the same task. The unstructured mesh more uniformly distributes data samples over mesh cells to improve the accuracy of the reconstructed signal. By empirically balancing bias and variance errors in selecting the number of unstructured cells as a function of the number of available samples, the method achieves asymptotic statistical convergence with known test data and reduces synthetic noise on experimental signals from Hall effect thrusters with greater success than the original grid-based strategy.

Title: Generative Modeling of Time-Dependent Densities via Optimal Transport and Projection Pursuit

Collaborators: Yunan Yang (Cornell University) and Romit Maulik (Pennsylvania State University & Argonne National Laboratory)

Abstract: Motivated by the computational difficulties incurred by popular deep learning algorithms for the generative modeling of temporal densities, we propose a cheap alternative which requires minimal hyperparameter tuning and scales favorably to high dimensional problems. In particular, we use a projection-based optimal transport solver [Meng et al., 2019] to join successive samples and subsequently use transport splines [Chewi et al., 2020] to interpolate the evolving density. When the sampling frequency is sufficiently high, the optimal maps are close to the identity and are thus computationally efficient to compute. Moreover, the training process is highly parallelizable as all optimal maps are independent and can thus be learned simultaneously. Finally, the approach is based solely on numerical linear algebra rather than minimizing a nonconvex objective function, allowing us to easily analyze and control the algorithm. We present several numerical experiments on both synthetic and real-world datasets to demonstrate the efficiency of our method. In particular, these experiments show that the proposed approach is highly competitive compared with state-of-the-art normalizing flows conditioned on time across a wide range of dimensionalities.

Title: An Introduction to the Theory of the Ergodic Partition (Undergraduate Honors Thesis)

Advisor: Ryan J. Alvarado (Amherst College)

Abstract: Evolving systems can rarely be studied through direct observation of their underlying state. More commonly, one is restricted to analyzing partial measurements of the system, known as observables. It is therefore important to determine the extent to which such observables recover useful information about the system in question. Towards this, we draw from techniques in ergodic theory, functional analysis, and probability theory to rigorously develop tools for identifying invariant regions in dynamical systems through the analysis of observables. We conclude by studying a computational approach to this problem, informed by the theory developed throughout the body of the thesis.

Title: Juggling Dynamics

Collaborator: Troy Shinbrot (Rutgers University)

Excerpt: How do jugglers with reaction times no better than 200 ms catch balls every 120 ms? In part, multitasking may allow multiple balls to be processed simultaneously, though how that is done with 11 balls—the Guinness world record—is far from clear. And in part, balls are not thrown to random locations, so each ball need not be tracked and caught independently. Indeed, up to five balls can be juggled while the juggler is blindfolded. Jugglers rely on making accurate throws and predictions of where the balls will travel. The accuracy required is a measure of how unstable and thus how difficult a particular juggling pattern is.