基于物理学的神经网络与混合的科尔摩戈罗夫-阿诺德网络和增强的拉格朗基函数用于解决部分微分方程
Zhaoyang Zhang1, Qingwang Wang2, Yinxing Zhang1
1Faculty of Information Engineering and Automation, Kunming University of Science and Technology, Kunming, 650500, China.
Scientific reports
|March 28, 2025
概括
使用Kolmogorov-Arnold网络 (KANs) 的物理信息神经网络 (PINNs) 改善了部分微分方程的解决. 通过动态学习约束,AL-PKAN模型提高了准确性和可解释性.
科学领域:
- 计算物理学的计算物理.
- 深度学习应用程序深度学习应用程序
- 数学建模的数学建模
背景情况:
- 基于物理学的神经网络 (PINNs) 对于解决部分微分方程 (PDEs) 至关重要.
- 在PINN中,传统的多层感知子 (MLP) 存在可解释性差和光谱偏差.
- 由于膨胀的惩罚因子,现有的PINN方法可能会面临优化问题.
研究的目的:
- 引入一种新的混合模型,AL-PKAN,灵感来自Kolmogorov-Arnold网络 (KANs),用于数学物理问题.
- 解决传统PINN中解释性和光谱偏差的局限性.
- 开发一个更强大的优化策略来处理PINNs中的约束.
主要方法:
- 一种混合编码器-解码器架构,结合了Gated Recurrent Unit (GRU) 和KAN模块.
- KAN将多变量函数分解为可训练的单变量 B-spline 激活函数,用于 spline 插值.
- 形成了一个增强的拉格朗基函数,使惩罚因子和拉格朗基乘数成为可学习的参数.
主要成果:
- AL-PKAN模型在基准实验中表现出了显著的准确性和通用性.
- 约束平衡的动态调制是通过可学习的参数实现的.
- 拟议的方法克服了光谱偏差,并提高了与MLP相比的解释性.
结论:
- AL-PKAN模型在基于物理学的神经网络中提供了一个有希望的进步.
- KAN显示了提高PINN的性能和可解释性的巨大潜力.
- 增强的拉格朗捷式方法为PINNs提供了一个更稳定,更适应性的优化框架.
相关概念视频
Ampere-Maxwell's Law: Problem-Solving
1.4K
A parallel-plate capacitor with capacitance C, whose plates have area A and separation distance d, is connected to a resistor R and a battery of voltage V. The current starts to flow at t = 0. What is the displacement current between the capacitor plates at time t? From the properties of the capacitor, what is the corresponding real current?
To solve the problem, we can use the equations from the analysis of an RC circuit and Maxwell's version of Ampère's law.
For the first part of the...
To solve the problem, we can use the equations from the analysis of an RC circuit and Maxwell's version of Ampère's law.
For the first part of the...
1.4K
Linear Approximation in Frequency Domain
502
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
502
Linear Approximation in Time Domain
460
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
460
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
438
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...
438
Modeling with Differential Equations
334
Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
334
Separable Differential Equations
365
A separable differential equation is a type of first-order differential equation where the derivative dy/dx can be expressed as a product of two functions: one that depends only on x and another that depends only on y. This allows for the rearrangement of the equation so that all terms involving y are on one side, and all terms involving x are on the other. This process, known as the separation of variables, simplifies the process of solving the equation by enabling the integration of both...
365


