通过神经ODE方法对非线性PDEs进行强大的时刻识别
Shaoxuan Chen1, Su Yang1, Panayotis G Kevrekidis1,2,3
1Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, Massachusetts 01003-4515, USA.
Chaos (Woodbury, N.Y.)
|December 19, 2025
概括
本研究介绍了一个神经普通微分方程 (神经ODE) 框架,用于从部分微分方程 (PDEs) 学习系统动态. 该方法在模拟稀疏,杂的数据方面表现出色,提供了强大的减少顺序时刻动态发现.
科学领域:
- 计算物理 计算物理
- 应用数学 应用数学 应用数学
- 机器学习 机器学习
背景情况:
- 部分微分方程 (PDEs) 控制复杂的物理系统.
- 从数据中学习动态的传统方法往往需要密集,干净的观察.
- 降低级建模对于高维系统的高效模拟和分析至关重要.
研究的目的:
- 开发一个数据驱动的框架,从PDE-governed系统中学习减少顺序的动态.
- 为了从稀疏,不规则和杂的时间序列数据中实现强大的动态建模.
- 发现复杂系统的可解释的低维表示.
主要方法:
- 利用神经常规微分方程 (神经ODE) 直接建模时刻轨迹.
- 在缺乏分析闭包的系统中,使用Stiefel多元组优化用于数据驱动的坐标转换.
- 将框架应用于非线性施罗丁格和费舍尔-科尔莫戈罗夫-彼得罗夫斯基-皮斯库诺夫反应扩散系统.
主要成果:
- 神经ODE框架准确地从有限,不规则和杂的数据中恢复时刻动态.
- 成功发现了没有分析闭包的系统的低维表示中的闭合时刻动力学.
- 在数据有限的场景中,与基于物理的模型相比,证明了更高的外推精度.
结论:
- 拟议的神经ODE框架提供了一种强大而灵活的方法来学习可解释的,低维的时刻动态.
- 该方法在数据有限的环境中显示出显著的稳定性,优于传统技术.
- 能够对复杂的PDE控制系统进行可靠的建模和分析,即使有不完整的观测.
相关概念视频
01:26Linear Approximation in Frequency Domain
329
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....
329
01:21Linear Approximation in Time Domain
318
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,...
318
01:18Second Order systems II
369
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
369
01:29Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
261
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...
261
01:26Transmission-Line Differential Equations
935
Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured from...
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured from...
935
01:24Difference Equation Solution using z-Transform
594
The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
The z-transform facilitates handling delayed signals by shifting the signal in the z-domain, which corresponds to delaying the signal in the time domain, and advancing signals by similarly shifting in the...
594