训练硬神经普通微分方程与明确的理性泰勒数列方法
Colby Fronk1, Linda Petzold2,3
1Department of Chemical Engineering, University of California, Santa Barbara, Santa Barbara, California 93106, USA.
Chaos (Woodbury, N.Y.)
|July 18, 2025
概括
我们开发了新的明确的理性泰勒方法来训练硬神经普通微分方程. 这些方法提供了高效率和稳定性,降低了复杂动态建模的计算成本.
科学领域:
- 数字分析 数字分析
- 机器学习 机器学习
- 动态系统 动态系统
背景情况:
- 刚性神经普通微分方程 (NODE) 提出了重要的计算挑战.
- 对于刚性NODE的传统隐式方法通常是计算密集的.
- 在数据驱动模拟中,对刚性动态的高效稳定训练至关重要.
研究的目的:
- 引入新的显式理性泰勒序列方法,用于直接训练刚性NODE.
- 提高学习刚性动态系统的效率和数值稳定性.
- 为传统隐式方法提供一个计算缩小的替代方案.
主要方法:
- 开发二级和三级显式理性泰勒数列方案.
- 对拟议的明确方案的A-稳定性的证明.
- 应用这些方法来训练硬系统,包括范德波尔振荡器.
主要成果:
- 显式方案实现高效率与一个单一的线性解决每一个时间步骤.
- 提出的方法表现出强大的数值稳定性,即使在大型步骤大小.
- 在没有隐式方案常见的稳定性问题的情况下,证明了对刚性动态的有效学习.
- 与传统隐式方法相比,显著降低了计算成本.
结论:
- 显式理性泰勒方法为训练硬NODE提供了一种高效和稳定的方法.
- 这些方法扩大了对复杂动态的数据驱动模拟的能力.
- 这些发现支持在基于网格的模拟和基于物理的神经网络中的应用.
相关概念视频
01:26Linear Approximation in Frequency Domain
136
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....
136
01:21Linear Approximation in Time Domain
125
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,...
125
01:18Second Order systems II
173
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.
173
01:29Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
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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...
101
01:26Transmission-Line Differential Equations
405
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...
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...
405
01:20Discrete-Time Fourier Series
368
The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...
For a discrete-time periodic signal x[n]...
368