Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures

Abstract

Parameter identification determines the essential system parameters required to build real-world dynamical systems by fusing crucial physical relationships and experimental data. However, the data-driven approach faces main difficulties, such as a lack of observational data, discontinuous or inconsistent time trajectories, and noisy measurements. The ill-posedness of the inverse problem comes from the chaotic divergence of the forward dynamics. Motivated by the challenges, we shift from the Lagrangian particle perspective to the state space flow field's Eulerian description. Instead of using pure time trajectories as the inference data, we treat statistics accumulated from the Direct Numerical Simulation (DNS) as the observable, whose continuous analog is the steady-state probability density function (PDF) of the corresponding Fokker-Planck equation (FPE). We reformulate the original parameter identification problem as a data-fitting, PDE-constrained optimization problem. An upwind scheme based on the finite-volume method that enforces mass conservation and positivity preserving is used to discretize the forward problem. We present theoretical regularity analysis for evaluating gradients of optimal transport costs and introduce three different formulations for efficient gradient calculation. Numerical results using the quadratic Wasserstein metric from optimal transport demonstrate this novel approach's robustness for chaotic dynamical system parameter identification.

RNA velocity unraveled

“…i.e., the expectation of P(x;t) under the sampling law. P(x) is called the occupation measure of the process, and reports the probability that a trajectory is observed to be in state x, a slight generalization of the usual definition [141][142][143].…”

RNA velocity unraveled

“…i.e., the expectation of P (x, t) under the sampling law. P (x) is called the occupation measure of the process, and reports the probability that a trajectory is observed to be in state x, a slight generalization of the usual definition [139][140][141].…”

Learning Dynamical Systems From Invariant Measures

“…In [21,65], an Eulerian perspective was adopted to handle such difficulties, and velocity models were constructed to yield the same asymptotic statistics as the observed measurements, rather than seeking a pointwise match with time trajectories or Lagrangian velocities. More specifically, instead of directly treating the noisy observations {x η (t i )} n i=1 of an autonomous flow ẋ = v * (x) as inference data, the approaches in [21,65] considered the occupation measure ρ * , where for each measurable set B,…”

“…When the size of θ is large, it is practical to use gradient-based optimization methods for solving the optimization problem (2), and one has to compute the essential gradient ∂ θ J . In [65], this was handled by viewing ρ ε (v(θ)) as the dominant eigenvector of a regularized Markov matrix originating from an upwind finite volume discretization of the continuity equation. The derivative ∂ θ J was then seamlessly computed via the adjoint-state method [65].…”

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We extend the methodology in [65] to learn autonomous continuous-time dynamical systems from invariant measures. We assume that our data accurately describes the dynamics' asymptotic statistics but that the available time history of observations is insufficient for approximating the Lagrangian velocity.

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