Critical Sampling for Robust Evolution Operator Learning of Unknown Dynamical Systems

TL;DR

This paper tackles the data-efficiency problem of learning evolution operators for unknown dynamical systems. It introduces a forward-backward multi-step reciprocal prediction framework to estimate modeling error without ground truth and uses this to adaptively select high-error samples. A joint spatial-temporal network augments sparse data by learning local spatial dynamics and enforcing cross-domain consistency, significantly reducing the required samples while maintaining accuracy. Theoretical analysis links reciprocal prediction error to network error, and extensive experiments across ODEs and a PDE demonstrate substantial practical gains in sample efficiency and long-term predictive performance. The approach promises practical impact for data-limited real-world dynamical systems by lowering data collection costs and enabling robust long-horizon predictions.

Abstract

Given an unknown dynamical system, what is the minimum number of samples needed for effective learning of its governing laws and accurate prediction of its future evolution behavior, and how to select these critical samples? In this work, we propose to explore this problem based on a design approach. Starting from a small initial set of samples, we adaptively discover critical samples to achieve increasingly accurate learning of the system evolution. One central challenge here is that we do not know the network modeling error since the ground-truth system state is unknown, which is however needed for critical sampling. To address this challenge, we introduce a multi-step reciprocal prediction network where forward and backward evolution networks are designed to learn the temporal evolution behavior in the forward and backward time directions, respectively. Very interestingly, we find that the desired network modeling error is highly correlated with the multi-step reciprocal prediction error, which can be directly computed from the current system state. This allows us to perform a dynamic selection of critical samples from regions with high network modeling errors for dynamical systems. Additionally, a joint spatial-temporal evolution network is introduced which incorporates spatial dynamics modeling into the temporal evolution prediction for robust learning of the system evolution operator with few samples. Our extensive experimental results demonstrate that our proposed method is able to dramatically reduce the number of samples needed for effective learning and accurate prediction of evolution behaviors of unknown dynamical systems by up to hundreds of times.

Figures (9)

• Figure 1: Illustration of the proposed method of critical sampling for accurately learning the evolution behaviors of unknown dynamical systems.
• Figure 2: Illustration of the proposed idea of multi-step reciprocal prediction error. The left figure shows the robust evolution prediction on locations with small multi-step reciprocal prediction errors, where the forward and backward paths align perfectly. The right figure shows the inaccurate evolution prediction on locations with large multi-step reciprocal prediction errors, where we need to select more critical samples.
• Figure 3: Examples of multi-step reciprocal prediction errors on Damped Pendulum and 2D Nonlinear ODE systems. After training the forward and backward evolution networks with 225 samples on the Damped Pendulum system and 467 samples on the 2D Nonlinear system, we take the locations with large/small multi-step reciprocal prediction error as the starting point in these example trajectories. Here, we take the reciprocal step as $K=5$.
• Figure 4: Correlation between network modeling error and multi-step reciprocal prediction error on Damped Pendulum and 2D Nonlinear ODE systems.
• Figure 5: The critical sampling and adaptive learning results on four dynamical systems.

Theorems & Definitions (8)

• Lemma III.1
• proof
• Lemma III.2
• proof
• Lemma III.3
• proof
• Theorem III.4
• proof