Data-driven computation for periodic stochastic differential equations

TL;DR

This work addresses stochastic systems under time-periodic forcing by extending data-driven Fokker-Planck methods to the periodic setting, enabling computation of the time-periodic invariant density $u(x,t)$ that satisfies the periodic equation Lu = 0 with $u(x,t+T)=u(x,t)$. It develops a grid-based finite-difference solver and a mesh-free neural-network solver, each enforcing time periodicity and leveraging Monte Carlo sampling to recover the periodic density; convergence speed to the periodic measure is analyzed via a Floquet-type decomposition of coupling times. The authors propose and validate a coupling-based estimator for the geometric ergodicity rate $r$ and a periodic pre-factor $C(t)$, demonstrated on the Stuart–Landau oscillator, a chaotic forced Van der Pol system, and periodically driven neuronal oscillators. This framework advances data-driven analysis of periodic stochastic dynamics and provides practical tools for quantifying convergence to periodic equilibria in applications such as neuroscience and coupled oscillatory networks.

Abstract

Many stochastic differential equations in various applications like coupled neuronal oscillators are driven by time-periodic forces. In this paper, we extend several data-driven computational tools from autonomous Fokker-Planck equation to the time-periodic setting. This allows us to efficiently compute the time-periodic invariant probability measure using either grid-base method or artificial neural network solver, and estimate the speed of convergence towards the time-periodic invariant probability measure. We analyze the convergence of our algorithms and test their performances with several numerical examples.

Figures (15)

• Figure 1: principal angles between $\operatorname{Ker}\mathbf{A}$ and $\Theta_D$ for $D=1,2,3$. First row: Example 1; second row: Example 2; third row: Example 3. Left: Forward Euler method; middle: Backward Euler method; right: Crank-Nicolson method.
• Figure 2: Implicit scheme for computing the principal angles between $\operatorname{Ker}\mathbf{A}$ and $\Theta_D$ for $D=1,2,3$. Left: $t$ in whole time interval. Right: $t$ in part time interval.
• Figure 3: Explicit scheme for computing the principal angles between $\operatorname{Ker}\mathbf{A}$ and $\Theta_D$ for $D=1,2,3$. Left: $t$ in whole time interval. Right: $t$ in part time interval.
• Figure 4: Crank-Nicolson scheme for computing the principal angles between $\operatorname{Ker}\mathbf{A}$ and $\Theta_D$ for $D=1,2,3$. Left: $t$ in whole time interval. Right: $t$ in part time interval.
• Figure 5: Empirical spatial distribution of error term for ring density function as in Section 6.1.

Theorems & Definitions (12)

• Definition 2.1: Periodic Probability Measure
• Definition 2.2: Periodic Ergodicity
• Definition 2.3: Periodic Contraction
• Definition 2.4: Coupling of Probability Measure
• Definition 2.5: Coupling of Stochastic Process
• Theorem 3.1
• proof
• Theorem 5.1
• Lemma 5.2
• Proposition 5.1