Minimum Reduced-Order Models via Causal Inference

TL;DR

This paper introduces a causation entropy-based method for constructing sparse reduced-order models of high-dimensional dynamical systems, demonstrating efficiency and effectiveness in capturing complex dynamics and unobserved behaviors.

Contribution

It presents a novel approach using causation entropy to identify important features for sparse ROMs, including Gaussian approximations that handle non-Gaussian data efficiently.

Findings

Causation entropy effectively ranks feature importance for ROM construction.

Gaussian approximation of causation entropy performs well with non-Gaussian data.

ROMs built with this method accurately recover long-term statistics and unobserved dynamics.

Abstract

Constructing sparse, effective reduced-order models (ROMs) for high-dimensional dynamical data is an active area of research in applied sciences. In this work, we study an efficient approach to identifying such sparse ROMs using an information-theoretic indicator called causation entropy. Given a feature library of possible building block terms for the sought ROMs, the causation entropy ranks the importance of each term to the dynamics conveyed by the training data before a parameter estimation procedure is performed. It thus allows for an efficient construction of a hierarchy of ROMs with varying degrees of sparsity to effectively handle different tasks. This article examines the ability of the causation entropy to identify skillful sparse ROMs when a relatively high-dimensional ROM is required to emulate the dynamics conveyed by the training dataset. We demonstrate that a Gaussian approximation of the causation entropy still performs exceptionally well even in presence of highly non-Gaussian statistics. Such approximations provide an efficient way to access the otherwise hard to compute causation entropies when the selected feature library contains a large number of candidate functions. Besides recovering long-term statistics, we also demonstrate good performance of the obtained ROMs in recovering unobserved dynamics via data assimilation with partial observations, a test that has not been done before for causation-based ROMs of partial differential equations. The paradigmatic Kuramoto-Sivashinsky equation placed in a chaotic regime with highly skewed, multimodal statistics is utilized for these purposes.