误差估计和基于物理的增强神经网络,用于热合不可压缩的纳维埃·斯托克斯方程
Shoaib Goraya1, Nahil Sobh2, Arif Masud1
1Department of Civil and Environmental Engineering, University of Illinois at Urbana Champaign, Urbana, IL 61801, USA.
概括
物理信息神经网络 (PINNs) 为解决部分微分方程 (PDEs) 提供了一个有前途的方法. 本研究提供了PINNs应用于复杂的流体动力学问题的趋同分析和误差估计,提高了它们的可靠性.
科学领域:
- 计算流体动力学 计算流体动力学
- 数字分析 数字分析
- 机器学习用于科学.
背景情况:
- 物理信息神经网络 (PINNs) 正在成为接近部分微分方程 (PDEs) 解决方案的强大工具.
- 然而,对它们的误差估计和收性质的严格分析是有限的,这阻碍了对它们的经验成功的充分理解.
- 这种差距在复杂的多物理问题中尤为明显.
研究的目的:
- 为了呈现一个全面的融合分析和错误估计PINNs应用到热合不可压缩的纳维埃-斯托克斯方程.
- 研究PINN中的训练错误和泛化错误之间的关系.
- 为准确的PINN预测提供参数选择的实际指导.
主要方法:
- 开发和应用PINNs来解决热合不可压缩的纳维埃-斯托克斯方程.
- 使用贝尔特拉米流量模型问题进行初始分析.
- 以后对培训残留和配合点的误差估计的推导.
- 将压力稳定项 (压力波松方程) 纳入PINN框架.
- 将PINN结果与稳定有限元素方法 (FEM) 的比较.
主要成果:
- 证明PINNs中的小训练错误会导致小的概括错误.
- 针对培训剩余点和拼接点的总错误,以后确定了对整体错误的收率.
- 表明添加一个压力稳定项显著提高压力场的精度 (一个数量级).
- 与稳定的FEM相比,验证了PINN的性能,突出了PINNs的有效性.
结论:
- 该研究为PINNs在解决复杂的PDEs,特别是流体动力学中的可靠应用提供了关键的理论依据.
- 由此得出的误差估计和趋同率为优化PINN培训和预测提供了实际指导方针.
- PINNs,特别是与压力稳定等基于物理学的增强功能,显示出作为传统数值方法的准确和高效替代品的重大承诺.
相关概念视频
Navier–Stokes Equations
577
For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
577
Newtonian Fluid: Problem Solving
259
Newtonian fluids exhibit a constant viscosity, meaning their shear stress and shear strain rate are directly proportional. This property ensures a predictable and stable response to applied forces, maintaining a linear relationship between force and flow. Examples include water, air, and light oils, consistently demonstrating this proportional behavior regardless of external conditions.
A velocity gradient forms within the fluid when a Newtonian fluid is placed between two parallel plates, with...
A velocity gradient forms within the fluid when a Newtonian fluid is placed between two parallel plates, with...
259
Maxwell-Boltzmann Distribution: Problem Solving
1.6K
Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
1.6K
Estimation of the Physical Quantities
4.4K
On many occasions, physicists, other scientists, and engineers need to make estimates of a particular quantity. These are sometimes referred to as guesstimates, order-of-magnitude approximations, back-of-the-envelope calculations, or Fermi calculations. The physicist Enrico Fermi was famous for his ability to estimate various kinds of data with surprising precision. Estimating does not mean guessing a number or a formula at random. Instead, estimation means using prior experience and sound...
4.4K
Thermal expansion and Thermal stress: Problem Solving
1.2K
San Francisco's Golden Gate Bridge is exposed to temperatures ranging from -15 °C to 40 °C. At its coldest, the main span of the bridge is 1275 m long. Assuming that the bridge is made entirely of steel, what is the change in its length between these temperatures?
To solve the problem, first, identify the known and unknown quantities. The initial length (L) of the bridge is 1275 m, the coefficient of linear expansion (α) for steel is 12 x 10-6/°C, and the change in...
To solve the problem, first, identify the known and unknown quantities. The initial length (L) of the bridge is 1275 m, the coefficient of linear expansion (α) for steel is 12 x 10-6/°C, and the change in...
1.2K
Accelerating Fluids
1.1K
When a fluid is in constant acceleration, the pressure and buoyant force equations are modified. Suppose a beaker is placed in an elevator accelerating upward with a constant acceleration, a. In the beaker, assume there is a thin cylinder of height h with an infinitesimal cross-sectional area, ΔS.
The motion of the liquid within this infinitesimal cylinder is considered to obtain the pressure difference. Three vertical forces act on this liquid:
The motion of the liquid within this infinitesimal cylinder is considered to obtain the pressure difference. Three vertical forces act on this liquid:
1.1K


