[论文解读] Inverse problems for history-enriched linear model reduction

Arjun Vijaywargiya, George Biros|arXiv (Cornell University)|Jan 11, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

The paper derives exact history-enriched reduced models via Mori-Zwanzig for linear driven systems and formulates inverse problems to learn the memory and noise operators from data, with a greedy time-marching solution and well-posedness analysis.

ABSTRACT

Standard projection-based model reduction for dynamical systems incurs closure error because it only accounts for instantaneous dependence on the resolved state. From the Mori-Zwanzig (MZ) perspective, projecting the full dynamics onto a low-dimensional resolved subspace induces additional noise and memory terms arising from the dynamics of the unresolved component in the orthogonal complement. The memory term makes the resolved dynamics explicitly history dependent. In this work, based on the MZ identity, we derive exact, history-enriched models for the resolved dynamics of linear driven dynamical systems and formulate inverse problems to learn model operators from discrete snapshot data via least-squares regression. We propose a greedy time-marching scheme to solve the inverse problems efficiently and analyze operator identifiability under full and partial observation data availability. For full observation data, we show that, under mild assumptions, the operators are identifiable even when the full-state dynamics are governed by a general time-varying linear operator, whereas with partial observation data the inverse problem has a unique solution only when the full-state operator is time-invariant. To address the resulting non-uniqueness in the time-varying case, we introduce a time-smoothing Tikhonov regularization. Numerical results demonstrate that the operators can be faithfully reconstructed from both full and partial observation data and that the learned history-enriched MZ models yield accurate trajectories of the resolved state.

研究动机与目标

• Motivate closure modeling for reduced-order dynamics by accounting for memory and noise from unresolved modes.
• Formulate inverse problems to learn the Mori-Zwanzing operators from snapshot data.
• Develop a greedy time-marching algorithm to efficiently solve the inverse problems.
• Analyze identifiability of the operators under full and partial observation data.
• Propose regularization to stabilize ill-posed cases under partial data.

提出的方法

• Derive the Mori-Zwanzig identity for linear dynamical systems to obtain a Volterra integro-differential equation for the resolved variables.
• Formulate continuous and discrete inverse problems to recover the Markovian operator R, memory kernel K, and noise operator B from snapshot data.
• Propose a greedy time-marching least-squares scheme that solves for the kernels step-by-step due to the causal, lower-triangular structure of the problem.
• Provide a stationarity (time-translation invariance) assumption to simplify kernels to depend on time lag t-s, facilitating tractable identification.
• Analyze well-posedness: full data problems are well-posed; partial data problems are well-posed only for time-invariant A; introduce time-smoothing regularization for ill-posed cases.
• Discuss finite-memory approximations and non-stationary kernel considerations and their impact on identifiability.

Figure 1 : (a) True and predicted trajectories for two degrees of freedom (selected as the resolved variables) for a one-dimensional advection equation with periodic boundary conditions. Predictions are shown for a data-driven Markovian model and a data-driven MZ model ( 1.2 ). (b) Absolute error in

实验结果

研究问题

• RQ1 Can the Markovian, memory, and noise operators in the Mori-Zwanzig reduced model be uniquely identified from snapshot data under full observation?
• RQ2 What are the identifiability and stability conditions for these operators when only partial state observations are available?
• RQ3 How does time-varying versus time-invariant full-state operators A affect the solvability of the inverse problems?
• RQ4 Does a greedy time-marching approach yield a practical solution with provable properties for the inverse problems?
• RQ5 What regularization strategies stabilize reconstruction in ill-posed partial-data scenarios?

主要发现

• The operators R, B, and K are uniquely recoverable under time-invariant A from both full and partial data.
• For time-varying A, R and K are uniquely recoverable with full data, while partial data yields ill-posedness unless A is time-invariant; a time-smoothing regularization can restore stability.
• A greedy time-marching scheme enables efficient reconstruction by solving decoupled or sequential least-squares problems at each time step.
• Numerical experiments show faithful reconstruction of MZ operators from both full and partial data and accurate trajectory prediction with history-enriched models.
• The learned history-enriched MZ models outperform purely Markovian models in capturing resolved dynamics in test scenarios.

Figure 3 : Predicted trajectories of the resolved variables in all the reaction-diffusion-advection test cases for a single test initial condition. The trajectories are obtained by solving ( 1.2 ), with operators $\mathord{\btensor R}$ , $\mathord{\btensor K}$ , and $\mathord{\btensor B}$ reconstruc