通过全协方差高斯混合网络识别混合噪声随机系统的参数耦合与不确定性

中文摘要 AI 辅助

由混合噪声驱动的随机动力系统的参数识别颇具挑战性，原因在于其似然函数难以处理。我们提出PENN-GMD，这是一种参数估计神经网络，可将部分观测轨迹映射到系统参数上的高斯混合分布（GMD）。与传统不确定性估计不同，该GMD采用全协方差矩阵，以明确揭示参数耦合与多模态似然结构。该网络通过满射参数化最小化负对数似然进行训练，该参数化对所有GMD约束进行硬编码，从而近似真实似然。我们在五个复杂度递增的数值示例上验证了该方法，包括由分形高斯噪声和莱维噪声驱动的系统、带有色噪声的振荡器、不同可观测性下的耦合神经元，以及存在不可识别随机扰动的气动弹性翼型。结果表明，PENN-GMD可准确恢复似然分布、捕捉参数耦合，并通过方差扩大或模式分裂自然诊断不可识别性。这些能力使PENN-GMD成为复杂随机系统中不确定性感知参数识别的实用工具，而传统基于似然的方法在这类系统中不可行。

英文摘要

Parameter identification of stochastic dynamical systems driven by mixed noises is challenging due to intractable likelihood functions. We propose PENN-GMD, a parameter estimation neural network that maps partially observed trajectories to a Gaussian mixture distribution (GMD) over the system parameters. Unlike conventional uncertainty estimates, the GMD employs full covariance matrices to explicitly reveal parameter couplings and multi-modal likelihood structures. The network is trained by minimizing the negative log-likelihood via a surjective parameterization that hard-encodes all GMD constraints, thereby approximating the true likelihood. We validate the method on five numerical examples with increasing complexity, including systems driven by fractional Gaussian and Lévy noises, oscillators with colored noise, coupled neurons under different observability, and an aeroelastic airfoil with unidentifiable stochastic disturbances. Results demonstrate that PENN-GMD accurately recovers likelihood distributions, captures parameter couplings, and naturally diagnoses non-identifiability through variance broadening or mode splitting. These capabilities establish PENN-GMD as a practical tool for uncertainty-aware parameter identification in complex stochastic systems where conventional likelihood-based methods are infeasible.

发表机构

• School of Science, Xi’an University of Posts and Telecommunications（西安邮电大学理学院）

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