通过优化离散损失来解决物理中的反向问题:在没有神经网络的情况下快速准确地学习
Petr Karnakov1, Sergey Litvinov1, Petros Koumoutsakos1
1Computational Science and Engineering Laboratory, Harvard John A. Paulson School of Engineering and Applied Sciences, Cambridge, MA 02138, USA.
PNAS nexus
|January 22, 2024
概括
这项研究引入了优化离散损失 (ODIL) 框架,该框架加速了与神经网络相比,以五个数量级加速解决由部分微分方程 (PDEs) 规范的反向问题. ODIL利用传统的PDE近似和机器学习工具来提高速度和准确性.
科学领域:
- 计算物理 计算物理
- 科学机器学习科学机器学习
- 数字分析 数字分析
背景情况:
- 物理学中的反向问题通常是通过部分微分方程 (PDEs) 建模的.
- 神经网络 (NN) 已被应用来解决这些反向问题,最大限度地减少基于PDE的损失函数.
- 现有的NN方法在计算速度和准确性方面存在局限性.
研究的目的:
- 引入一种新的框架,即优化离散损失 (ODIL),以显著加快PDE规范的反向问题的解决.
- 为了证明ODIL在计算速度,准确性和融合率方面优于物理信息神经网络 (PINNs).
- 为科学应用提供一个强大的工具,将数值方法和机器学习联系起来.
主要方法:
- 开发了优化离散损失 (ODIL) 框架,该框架使用 PDE 的离散近似值,而不是 NNs.
- 采用基于梯度和牛顿的方法来最小化离散成本函数.
- 集成的机器学习工具用于自动差异化和多网格技术来加速融合.
主要成果:
- 与基于NN的方法相比,在解决反向问题的过程中实现了五次级的加速.
- 与基于物理的神经网络相比,ODIL显示出更高的准确性和融合率.
- 成功地将ODIL应用于各种问题,包括PDE受约束优化,光流,系统识别,数据同化和纳维尔-斯托克斯方程.
结论:
- 在物理中,ODIL为解决反向问题的NN-based方法提供了一个计算效率高,准确的替代方案.
- 该框架继承了基于网格的PDE离散的理想特性,例如准确性和保存性.
- ODIL代表着一个重要的进步,将数值方法和机器学习与广泛的科学发现相结合.
相关概念视频
Ampere-Maxwell's Law: Problem-Solving
631
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...
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...
631
Fast Decoupled and DC Powerflow
195
The fast decoupled power flow method addresses contingencies in power system operations, such as generator outages or transmission line failures. This method provides quick power flow solutions, essential for real-time system adjustments. Fast decoupled power flow algorithms simplify the Jacobian matrix by neglecting certain elements, leading to two sets of decoupled equations:
195
Principle of Linear Impulse and Momentum for a Single Particle: Problem Solving
203
Consider a wooden box and a cylinder of known masses m1 and m2, respectively, hanging from a ceiling with the help of a massless pulley system.
203
Conservation of Momentum: Problem Solving
10.4K
Solving problems using the conservation of momentum requires four basic steps:
10.4K
Solving Problems in Physics
5.8K
Problem-solving is the ability to apply general physical principles to specific situations, usually expressed by equations. It is an essential skill in physics, and can also be useful for applying physics in everyday life as well. Analytical skills and problem-solving abilities can be applied to new situations, compared to a list of facts, which can never be extensive enough to include every possible circumstance. To solve physics problems, a certain amount of creativity and insight is...
5.8K
Bernoulli's Equation: Problem Solving
1.3K
A Venturi meter is essential for measuring fluid flow rates in pipelines. It utilizes the relationship between fluid velocity and pressure described by Bernoulli's equation. When installed in a sewage system, the Venturi meter accurately determines the wastewater flow rate by measuring pressure differences.
The first step is to compute the cross-sectional areas of the pipe and the Venturi throat to analyze the pressure difference indicated by the pressure gauge. Next, the continuity...
The first step is to compute the cross-sectional areas of the pipe and the Venturi throat to analyze the pressure difference indicated by the pressure gauge. Next, the continuity...
1.3K


