Data-driven optimal prediction with control

TL;DR

The paper tackles predicting averaged trajectories in dynamical systems with unresolved variables under control. It fuses Mori-Zwanzig optimal prediction with Dynamic Mode Decomposition to form OPc, a data-driven method that learns a finite-dimensional operator $\mathbf{A}_{opc}$ and a memory-correction term to predict $\mathbf{x}(t)$ from a single trajectory, even when the control operator $\mathcal{B}$ is unknown. OPc yields a locally optimal linear operator whose spectrum informs stability and can be computed much faster than Monte Carlo projections, making it practical for real-time or many-query settings. Across Hamiltonian test problems with constant and damped controls, OPc accurately tracks the averaged dynamics and outperforms DMDc in handling unresolved variables, while accommodating cases with known or unknown control influence.

Abstract

This study presents the extension of the data-driven optimal prediction approach to the dynamical system with control. The optimal prediction is used to analyze dynamical systems in which the states consist of resolved and unresolved variables. The latter variables can not be measured explicitly. They may have smaller amplitudes and affect the resolved variables that can be measured. The optimal prediction approach recovers the averaged trajectories of the resolved variables by computing conditional expectations, while the distribution of the unresolved variables is assumed to be known. We consider such dynamical systems and introduce their additional control functions. To predict the targeted trajectories numerically, we develop a data-driven method based on the dynamic mode decomposition. The proposed approach takes the $\mathit{measured}$ trajectories of the resolved variables, constructs an approximate linear operator from the Mori-Zwanzig decomposition, and reconstructs the $\mathit{averaged}$ trajectories of the same variables. It is demonstrated that the method is much faster than the Monte Carlo simulations and it provides a reliable prediction. We experimentally confirm the efficacy of the proposed method for two Hamiltonian dynamical systems.

Figures (6)

• Figure 1: Comparison of trajectories corresponding to the dynamical system \ref{['eq::test_problem1']} and the control function $\mathbf{g}_c(\mathbf{y}) = [0.1, 0.1, -0.01, 0.01]$. (a): trajectories of resolved and unresolved variables corresponding to a single initialization. (b) and (c): dynamics obtained from the MC projection method and OPc method have a similar trend. This trend shows the stability of averaged trajectories corresponding to the resolved variables. A straightforward DMDc constructed from a single measurement shows regular trajectories, contradicting the ground-truth decay trajectories via the MC projection and OPc methods.
• Figure 2: Comparison of the trajectories corresponding to the dynamical system \ref{['eq::test_problem1']} and the control function $\mathbf{g}_d(\mathbf{y}) = -0.01 \mathbf{y}$. (a): the trajectories of resolved and unresolved variables generated after a single initialization. (b) and (c): the proposed optimal prediction with control (OPc) method captures the same dynamics trend as the ground-truth solution computed via Monte Carlo (MC) projection. DMDc shows a significantly slower decay of the trajectories and does not provide an accurate approximation of the ground-truth averaged trajectories.
• Figure 3: Comparison of trajectories for the constant control function $\mathbf{g}(\mathbf{y}) = \mathbf{c}$ and dynamical system \ref{['eq::test_problem_2']}. (a): trajectories of resolved and unresolved variables with a single initialization. (b) and (c): comparison of trajectories computed with MC projection, DMDc, and OPc methods for the resolved variables. The OPc method captures the trend of ground-truth averaged trajectories given by MC projection. The DMDc method gives stable trajectories that differ from the observed decay behavior of the averaged trajectories from both OPc and MCc projection methods.
• Figure 4: Comparison of trajectories for the damped control function $\mathbf{g}_d(\mathbf{y}) = -0.01 \mathbf{y}$ and dynamical system \ref{['eq::test_problem_2']}. (a): trajectories of resolved and unresolved variables with a single initialization. (b) and (c): comparison of trajectories computed with MC projection, OPc, and DMDc methods for the resolved variables. The OPc method captures the trend of the ground-truth averaged trajectories given by MC projection. DMDc gives slow decay trajectories that significantly differ from the expected trajectories.
• Figure 5: Comparison of measured and averaged trajectories. (a): measurement trajectories show the decay of resolved and unresolved variables but with slightly different rates due to the values in the diagonal of matrix $\mathbf{B}_1$. (b) and (c): comparison of reconstructed average trajectories with the expected trajectories confirms that the proposed OPc method can recover matrix $\mathbf{B}_{opc}$. The latter corresponds to the given input vectors affecting resolved and unresolved variables.