2 research outputs found
Generative Modeling of Time-Dependent Densities via Optimal Transport and Projection Pursuit
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
Learning dynamics on invariant measures using PDE-constrained optimization
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.Comment: This article may be downloaded for personal use only. Any other use
requires prior permission of the author and AIP Publishing. This article
appeared in Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume
33, Issue 6, June 2023, and may be found at https://doi.org/10.1063/5.014967