Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty-猫眼课题宝

DOI：

10.48550/arxiv.2210.08095

发表时间：

2022-10

期刊：

ArXiv

影响因子：

通讯作者：

中科院分区：

其他

文献类型：

作者：

Luning Sun;D. Huang;Hao Sun-;Jian-Xun Wang

非线性动力学在科学和工程应用中无处不在，但大多数复杂系统的物理学还远未被完全理解。从测量数据中发现可解释的控制方程可以帮助我们理解和预测复杂动态系统的行为。尽管最近在该领域进行了大量工作，但从具有相当大噪声的非常稀疏的数据中稳健地提取显式模型形式仍然很棘手。此外，从噪声数据中量化和传播已识别系统的不确定性具有挑战性，相关文献仍然有限。为了弥补这一差距，我们开发了一种新颖的贝叶斯样条学习框架，以从具有量化不确定性的稀疏、噪声数据中识别非线性（空间）时间动力学的简约控制方程。该方法利用样条基础来处理数据稀缺和测量噪声，在此基础上可以准确计算一组导数以形成候选模型项库。方程残差用于以贝叶斯方式告知样条学习，其中采用近似贝叶斯不确定性校准技术来近似可训练参数的后验分布。为了促进稀疏性，开发了一种迭代顺序阈值贝叶斯学习方法，使用替代方向优化策略来系统地近似 L0 稀疏性约束。该算法在由规范常微分方程和偏微分方程控制的多个非线性动力系统上进行了评估，并通过与最先进的方法进行比较证明了该方法的优点/优越性。

Nonlinear dynamics are ubiquitous in science and engineering applications, but the physics of most complex systems is far from being fully understood. Discovering interpretable governing equations from measurement data can help us understand and predict the behavior of complex dynamic systems. Although extensive work has recently been done in this field, robustly distilling explicit model forms from very sparse data with considerable noise remains intractable. Moreover, quantifying and propagating the uncertainty of the identified system from noisy data is challenging, and relevant literature is still limited. To bridge this gap, we develop a novel Bayesian spline learning framework to identify parsimonious governing equations of nonlinear (spatio)temporal dynamics from sparse, noisy data with quantified uncertainty. The proposed method utilizes spline basis to handle the data scarcity and measurement noise, upon which a group of derivatives can be accurately computed to form a library of candidate model terms. The equation residuals are used to inform the spline learning in a Bayesian manner, where approximate Bayesian uncertainty calibration techniques are employed to approximate posterior distributions of the trainable parameters. To promote the sparsity, an iterative sequential-threshold Bayesian learning approach is developed, using the alternative direction optimization strategy to systematically approximate L0 sparsity constraints. The proposed algorithm is evaluated on multiple nonlinear dynamical systems governed by canonical ordinary and partial differential equations, and the merit/superiority of the proposed method is demonstrated by comparison with state-of-the-art methods.

Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty

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