学习使用符号回归方法对公民方程的近似符号解决方案
Benjamin G Cohen1, Burcu Beykal1, George M Bollas1
1Department of Chemical and Biomolecular Engineering, University of Connecticut, Storrs, CT 60629 USA.
概括
这项研究使用符号回归来发现没有数据的物理方程. 一个循序渐进的方法,结合领域知识,成功地模拟了扩散和 Burgers.
科学领域:
- * 计算物理 计算物理
- * 应用数学 * 应用数学
- * 机器学习 * 机器学习
背景情况:
- * 解决局部微分方程 (PDEs) 的传统方法通常需要大量数据或复杂的数值模拟.
- *符号回归提供了一个数据驱动的方法来发现治理方程,但可以与复杂的系统作斗争.
- * 整合领域知识可以引导符号回归向更高效和可解释的解决方案.
研究的目的:
- * 开发和演示一个逐步的符号回归策略来学习PDE的解决方案,而不需要先前的数据.
- * 调查将域名知识纳入的有效性,以简化搜索空间和改善模型发现.
- * 为扩散方程和Burgers方程生成可解释的符号解决方案.
主要方法:
- *采用了一种逐步符号回归方法,从学习系统物理学的部分模型开始.
- * 领域知识被利用来定义初始原始,并指导发现一个完整的物理模型.
- *该方法应用于扩散方程和不同对流系数的汉堡方程.
主要成果:
- * 该方法在扩散方程模型中获得了0.99的R平方值.
- * 汉堡方程的符号模型是通过不同对流系数的R平方值超过0.98来生成的.
- * 对伯格斯方程的发现解决方案被表示为扩散方程解决方案的变换.
结论:
- * 循序渐进的符号回归,加上领域知识,可以有效地学习动态系统的符号解决方案,没有数据.
- *这种方法提高了可解释性,并证明了专家直觉和自动发现之间的协同作用.
- *这些发现突出了开发更高效,更易理解的基于物理的机器学习模型的有希望的方向.
相关概念视频
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
292
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
292
Linearization and Approximation
18
Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
18
Implicit Differentiation: Problem Solving
29
Curves defined implicitly, where variables cannot be separated algebraically, require specialized techniques for analysis. The conchoid of Nicomedes exemplifies such a case. Its equation links x and y in a way that prevents isolation of one variable, making implicit differentiation essential to determine the slope and behavior at any point on the curve.The implicit form of the conchoid can be expressed as:To differentiate this equation, y is treated as a function of x, and the chain rule is...
29
Mathematical Modeling: Problem Solving
278
Mathematical modeling transforms real-world scenarios into mathematical expressions, allowing for structured problem-solving and analysis. This process involves defining the situation, assigning variables to measurable quantities, selecting an appropriate model, and solving the resulting equation. Such models are invaluable in finance, providing precise methods to evaluate investments, loans, and repayment structures.A widely used example is the calculation of fixed monthly payments on a loan,...
278
Gaussian Elimination: Problem Solving
162
Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...
162
Differential Equations: Problem Solving
25
When analyzing the motion of falling objects, it is essential to consider not only the force of gravity but also the opposing force of air resistance. A practical example involves releasing a heavy test weight during a safety check on a ship. As the weight falls from rest, gravity accelerates it downward while air resistance exerts an upward force that increases with velocity. This dynamic interplay of forces is well described by differential equations, which provide a mathematical framework...
25

