We propose a novel approach for performing dynamical system identification, based upon the comparison of simulated and observed physical invariant measures. While standard methods adopt a Lagrangian perspective by directly treating time-trajectories as inference data, we take on an Eulerian perspective and instead seek models fitting the observed global time-invariant statistics. With this change in perspective, we gain robustness against pervasive challenges in system identification including noise, chaos, and slow sampling. In the first half of this paper, we pose the system identification task as a partial differential equation (PDE) constrained optimization problem, in which synthetic stationary solutions of the Fokker-Planck equation, obtained as fixed points of a finite-volume discretization, are compared to physical invariant measures extracted from observed trajectory data. In the latter half of the paper, we improve upon this approach in two crucial directions. First, we develop a Galerkin-inspired modification to the finite-volume surrogate model, based on data-adaptive unstructured meshes and Monte-Carlo integration, enabling the approach to efficiently scale to high-dimensional problems. Second, we leverage Takens' seminal time-delay embedding theory to introduce a critical data-dependent coordinate transformation which can guarantee unique system identifiability from the invariant measure alone. This contribution resolves a major challenge of system identification through invariant measures, as systems exhibiting distinct transient behaviors may still share the same time-invariant statistics in their state-coordinates. Throughout, we present comprehensive numerical tests which highlight the effectiveness of our approach on a variety of challenging system identification tasks.

Invariant measures are widely used to compare chaotic dynamical systems, as they offer robustness to noisy data, uncertain initial conditions, and irregular sampling. However, large classes of systems with distinct transient dynamics can still exhibit the same asymptotic statistical behavior, which poses challenges when invariant measures alone are used to perform system identification. Motivated by Takens' seminal embedding theory, we propose studying invariant measures in time-delay coordinates, which exhibit enhanced sensitivity to the underlying dynamics. Our first result demonstrates that a single invariant measure in time-delay coordinates can be used to perform system identification up to a topological conjugacy. This result already surpasses the capabilities of invariant measures in the original state coordinate. Continuing to explore the power of delay-coordinates, we eliminate all ambiguity from the conjugacy relation by showing that unique system identification can be achieved using additional invariant measures in time-delay coordinates constructed from different observables. Our findings improve the effectiveness of invariant measures in system identification and broaden the scope of measure-theoretic approaches to modeling dynamical systems.

Such processes are visible in many applications ranging from geoscience, bioscience, and engineering to computer vision. In particular, deep learning algorithms, such as neural network parameterized normalizing flows, neural ordinary differential equations, diffusion models, and generative adversarial networks, have shown remarkable advances in learning and enabling rapid sampling from these stochastic processes. Such advances are further pronounced for very high-dimensional systems where classical methods are seen to saturate their effectiveness. However, the effective use of deep learning is frequently hampered by difficulties associated with computational cost as well as optimal hyperparameter selection. In this article, we propose a novel approach based on projection-pursuit optimal transport, which learns to sample from the densities of time-varying stochastic processes. It is competitive (both in terms of computational cost and accuracy) with a state-of-the-art deep learning algorithm (given by the neural spline flow). Crucially, our proposed method requires few hyperparameter choices by the user in contrast with most neural network-based methodologies. Thus, our main contributions to this work are as follows: 1. We implement a projection-pursuit optimal transport-based method to learn maps between time-varying densities from snapshots of particles sampled from these densities.