论文标题
贝叶斯学习在动态系统的管理方程中发现数学操作
Bayesian Learning to Discover Mathematical Operations in Governing Equations of Dynamic Systems
论文作者
论文摘要
从数据中发现管理方程对于多样化的科学学科至关重要，因为它们可以为动态系统的基本现象提供见解。这项工作通过设计具有深层神经网络式层次结构的数学操作网络（MathOnet）来提出一个用于管理方程式的新表示形式。具体而言，MathOnet分别由几层一层操作（例如SIN，COS，LOG）和二进制操作（例如， +， - ）堆叠。初始化的Mathonet通常被视为具有冗余结构的超级雕像，其子图可以产生管理方程。我们开发了一种稀疏的贝叶斯学习算法，通过在冗余数学操作上采用结构构造的先验来提取子图。通过演示混乱的洛伦兹系统，Lotka-Volterra系统和Kolmogorov-Petrovsky-Piskunov系统，该方法可以从有限的数学操作中发现了来自观察值的普通微分方程（ODE）和部分微分方程（ODE）和部分微分方程（PDE），而无需对ODES和PDES的可能进行任何先验的了解。
Discovering governing equations from data is critical for diverse scientific disciplines as they can provide insights into the underlying phenomenon of dynamic systems. This work presents a new representation for governing equations by designing the Mathematical Operation Network (MathONet) with a deep neural network-like hierarchical structure. Specifically, the MathONet is stacked by several layers of unary operations (e.g., sin, cos, log) and binary operations (e.g., +,-), respectively. An initialized MathONet is typically regarded as a super-graph with a redundant structure, a sub-graph of which can yield the governing equation. We develop a sparse group Bayesian learning algorithm to extract the sub-graph by employing structurally constructed priors over the redundant mathematical operations. By demonstrating the chaotic Lorenz system, Lotka-Volterra system, and Kolmogorov-Petrovsky-Piskunov system, the proposed method can discover the ordinary differential equations (ODEs) and partial differential equations (PDEs) from the observations given limited mathematical operations, without any prior knowledge on possible expressions of the ODEs and PDEs.
