Koopman-based Control for Stochastic Systems: Application to Enhanced Sampling

Abstract

We present a data-driven approach to use the Koopman generator for prediction and optimal control of control-affine stochastic systems. We provide a novel conceptual approach and a proof-of-principle for the determination of optimal control policies which accelerate the simulation of rare events in metastable stochastic systems.

Figures (5)

• Figure 1: Double-well potential $V$ and bias energies $B_i$ with $K_{\text{bias}}=4$ for $K_{\text{dw}}=1$ (a) and $K_{\text{dw}}=3$ (b)
• Figure 2: Absolute prediction errors (top row) and success rates $\delta$ (bottom row) with $K_{\text{bias}}=3$ (left) and $K_{\text{bias}}=4$ (right). Blue, orange and green lines stand for the cases when $K_{dw}=\{1,2,3\}$ respectively.
• Figure 3: Numerical results for the tracking optimal control problem (\ref{['track opt']}). (a)&(c) are the optimal signals and tracking errors for four parameter settings. Blue and orange lines stand for the systems when $K_{dw} \in \{1,3\}$, respectively. Panels (b)&(d) are the tracking performances for fixed $K_{\text{bias}}=3$ and $K_{\text{bias}}=4$, respectively.
• Figure 4: Numerical results of the problem (\ref{['cost_u']}) for the running cost (\ref{['running dw']}) with a range of parameters $c$. Left column are optimal signals and right column are expectations of the state $\mathbb{E}^{x}(X_t)$. From top to bottom are the results for the settings $(K_{\text{dw}},K_{\text{bias}}) \in \{(1,3),(3,3),(3,4)\}$, respectively, we omit the combination $(1, 4)$ as it is very similar to the first row. Solid and dashed lines stand for the reference and approximated expectations, respectively.
• Figure 5: Same as in Figure \ref{['fig:opt_dw']}, but for the running cost (\ref{['running bias']}).