Failure conditions for using interacting particle system data in EDMD approximations

Determine whether, and identify any conditions under which, computing extended dynamic mode decomposition (EDMD) estimates of projected Koopman or Perron–Frobenius operators using simulations of the interacting particle system with fully symmetric interactions (instead of data from the decoupled McKean–Vlasov stochastic differential equation) fails to yield accurate eigenvalue and eigenfunction estimates, given that only L2 convergence—not almost sure convergence—is available and that transfer operators for the interacting particle system are not well-defined in the authors’ framework.

Background

The paper develops a framework to define and estimate transfer operators (Koopman and Perron–Frobenius) for McKean–Vlasov equations by working with the decoupled McKean–Vlasov SDE, which preserves linearity and the Markov family property. This enables the use of Galerkin projections and EDMD for data-driven operator approximation and spectral analysis.

The authors note that while numerical experiments suggest that using data generated from the interacting particle system (IPS) can yield similar eigenvalue and eigenfunction estimates, the theoretical guarantees differ: only L2 convergence is available (not almost sure convergence), and, within their framework, transfer operators are not well-defined for the IPS. Consequently, it is explicitly unclear whether employing IPS data in place of decoupled SDE data fails under some conditions, motivating a concrete open question about characterizing such failure regimes.