This study analyzes the derivative-free loss method to solve a certain class
of elliptic PDEs using neural networks. The derivative-free loss method uses
the Feynman-Kac formulation, incorporating stochastic walkers and their
corresponding average values. We investigate the effect of the time interval
related to the Feynman-Kac formulation and the walker size in the context of
computational efficiency, trainability, and sampling errors. Our analysis shows
that the training loss bias is proportional to the time interval and the
spatial gradient of the neural network while inversely proportional to the
walker size. We also show that the time interval must be sufficiently long to
train the network. These analytic results tell that we can choose the walker
size as small as possible based on the optimal lower bound of the time
interval. We also provide numerical tests supporting our analysis.

该研究分析了使用神经网络解决一类椭圆型偏微分方程的无导数损失方法。该方法利用费曼 - 卡克公式，结合随机游走和其相应的平均值。我们研究了费曼 - 卡克公式中与时间间隔相关的效果，以及与计算效率、可训练性和采样误差的关系。我们的分析表明，训练损失的偏差与时间间隔和神经网络的空间梯度成正比，与游走大小成反比。我们还表明，时间间隔必须足够长以训练网络。这些分析结果告诉我们可以根据时间间隔的最优下限尽可能选择较小的游走大小。我们还提供数值测试来支持我们的分析。

求解偏微分方程之无导数损失方法之分析

An analysis of the derivative-free loss method for solving PDEs

Multiscale partial differential equations (PDEs) arise in various
applications, and several schemes have been developed to solve them
efficiently. Homogenization theory is a powerful methodology that eliminates
the small-scale dependence, resulting in simplified equations that are
computationally tractable. In the field of continuum mechanics, homogenization
is crucial for deriving constitutive laws that incorporate microscale physics
in order to formulate balance laws for the macroscopic quantities of interest.
However, obtaining homogenized constitutive laws is often challenging as they
do not in general have an analytic form and can exhibit phenomena not present
on the microscale. In response, data-driven learning of the constitutive law
has been proposed as appropriate for this task. However, a major challenge in
data-driven learning approaches for this problem has remained unexplored: the
impact of discontinuities and corner interfaces in the underlying material.
These discontinuities in the coefficients affect the smoothness of the
solutions of the underlying equations. Given the prevalence of discontinuous
materials in continuum mechanics applications, it is important to address the
challenge of learning in this context; in particular to develop underpinning
theory to establish the reliability of data-driven methods in this scientific
domain. The paper addresses this unexplored challenge by investigating the
learnability of homogenized constitutive laws for elliptic operators in the
presence of such complexities. Approximation theory is presented, and numerical
experiments are performed which validate the theory for the solution operator
defined by the cell-problem arising in homogenization for elliptic PDEs.

多尺度偏微分方程中的组分理论及椭圆 PDE 的合成定律的数据驱动学习方法。

椭圆算子同化学习

Learning Homogenization for Elliptic Operators

PDE learning is an emerging field that combines physics and machine learning
to recover unknown physical systems from experimental data. While deep learning
models traditionally require copious amounts of training data, recent PDE
learning techniques achieve spectacular results with limited data availability.
Still, these results are empirical. Our work provides theoretical guarantees on
the number of input-output training pairs required in PDE learning, explaining
why these methods can be data-efficient. Specifically, we exploit randomized
numerical linear algebra and PDE theory to derive a provably data-efficient
algorithm that recovers solution operators of 3D elliptic PDEs from
input-output data and achieves an exponential convergence rate with respect to
the size of the training dataset with an exceptionally high probability of
success.

本文介绍了一种结合了物理与机器学习的新兴领域：PDE 学习。我们提出了一种理论上保证数据效率的算法，可以从有限的输入输出数据中恢复 3D 椭圆型偏微分方程的解算子，并以极高的成功概率呈指数收敛率。