在非线性减少顺序建模中学习潜在动力学
Nicola Farenga1, Stefania Fresca1, Simone Brivio1
1MOX, Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milan, 20133, Italy.
概括
隐性动力学模型 (LDMs) 提供了一个新的数学框架,用于参数化的非线性时间依赖PDEs的减少顺序建模. 这种方法提高了复杂动态系统的精度和近似能力.
科学领域:
- 计算数学 计算数学 计算数学
- 科学机器学习科学机器学习
- 数字分析 数字分析
背景情况:
- 减少顺序建模 (ROM) 对于有效模拟参数化的非线性时间依赖部分微分方程 (PDEs) 至关重要.
- 现有的方法在准确捕捉这些系统的复杂动态和参数依赖性方面经常面临挑战.
- 对于高维度,时间依赖的问题,仍然需要强大而准确的近似技术.
研究的目的:
- 引入一个新的数学框架,隐性动力学模型 (LDMs),用于参数化的非线性时间依赖PDEs的减少顺序建模.
- 开发一种时间连续和可学习的方法,提供与完整订单模型 (FOM) 相比的边界近似误差.
- 通过基于深度学习的卷积架构来提高减少顺序模型的可解释性和准确性.
主要方法:
- 制定LDMs作为一个非线性维度缩小问题与一个受约束的潜在动态系统.
- 在连续时间设置中推导出误差和稳定性估计.
- 使用明确的Runge-Kutta方案开发一个时间离散公式 (ΔLDM) 和一个可学习的变体 (ΔLDMθ),采用深度神经网络 (DNN) 和具有亲属调制的卷积自编码器.
主要成果:
- 拟议的LDM框架提供了FOM解决方案的时间连续近似,使查询能够在任意时间实例中以受控的精度进行查询.
- 可学习的 ΔLDMθ 配方,利用 DNNs,实现相对于 FOM 的边界近似误差.
- 用伯格斯方程和向-扩散-反应方程进行的数值实验验验证了框架的准确性和处理参数化的PDEs的能力.
结论:
- 隐性动力学模型为复杂的,时间依赖的参数化PDEs的减少顺序建模提供了一个数学严格和有效的框架.
- 深度学习的整合,特别是卷积架构,增强了潜在表示的空间连贯性和可解释性.
- LDM方法显著提高了减少订单模型的准确性和近似能力,证明了在科学计算中的广泛适用性.
相关概念视频
01:21Linear Approximation in Time Domain
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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,...
60
01:26Linear Approximation in Frequency Domain
85
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....
85
01:18Second Order systems II
86
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.
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01:21First Order Systems
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First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...
83
01:23Linear time-invariant Systems
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A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
209
01:26Classification of Systems-I
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Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
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