用物理信息神经网络设计非线性观测器，提升状态估计精度与泛化能力。
KKL Observer Synthesis for Nonlinear Systems via Physics-Informed Learning
- 用物理信息网络学习正向映射，再用普通网络学逆映射。
- 在基准测试中表现优于现有方法，训练外泛化更强。
- 提供逼近误差的理论保证，适合高不确定性系统建模。
本文提出一种新型学习方法，用于设计自治非线性系统的Kazantzis-Kravaris或非线性卢恩伯格（KKL）观测器。设计KKL观测器需找到一个单射映射，将系统状态转换为高维观测器状态，其动态为线性且稳定；再通过逆映射将观测器状态映回原系统坐标以获得状态估计。但该变换及其逆变换的求解极为困难。本文采用物理信息神经网络学习正向映射，并用常规前馈神经网络学习逆映射。理论证明了状态估计对近似误差和系统不确定性的鲁棒性，包括将逼近质量与有限样本量关联的非渐近学习保证。通过基准算例的数值仿真验证了所提方法的有效性，在训练域外表现出更优的泛化能力，优于当前先进方法。
原文摘要 · Abstract (English)
This paper proposes a novel learning approach for designing Kazantzis-Kravaris or nonlinear Luenberger (KKL) observers for autonomous nonlinear systems. The design of a KKL observer involves finding an injective map that transforms the system state into a higher-dimensional observer state, whose dynamics is linear and stable. The observer's state is then mapped back to the original system coordinates via the inverse map to obtain the state estimate. However, finding this transformation and its inverse is quite challenging. We propose learning the forward mapping using a physics-informed neural network, and then learning its inverse mapping with a conventional feedforward neural network. Theoretical guarantees for the robustness of state estimation against approximation error and system uncertainties are provided, including non-asymptotic learning guarantees that link approximation quality to finite sample sizes. The effectiveness of the proposed approach is demonstrated through numerical simulations on benchmark examples, showing better generalization capability outside the training domain compared to state-of-the-art methods.
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