Identification of Differential Equations by Dynamics-Guided Weighted Weak Form with Voting

TL;DR

This paper introduces a dynamics-guided weighted weak form with voting strategy for more robust and efficient identification of differential equations from noisy data, leveraging adaptive test functions and ensemble voting.

Contribution

It proposes a novel adaptive weighting scheme for test functions based on dynamics indicators and a voting strategy to enhance robustness against noise in differential equation identification.

Findings

Effective identification of differential equations from noisy data.

Robustness demonstrated through systematic numerical experiments.

Improved accuracy over existing weak form methods.

Abstract

In the identification of differential equations from data, significant progresses have been made with the weak/integral formulation. In this paper, we explore the direction of finding more efficient and robust test functions adaptively given the observed data. While this is a difficult task, we propose weighting a collection of localized test functions for better identification of differential equations from a single trajectory of noisy observations on the differential equation. We find that using high dynamic regions is effective in finding the equation as well as the coefficients, and propose a dynamics indicator per differential term and weight the weak form accordingly. For stable identification against noise, we further introduce a voting strategy to identify the active features from an ensemble of recovered results by selecting the features that frequently occur in different weighting of test functions. Systematic numerical experiments are provided to demonstrate the robustness of our method.