We present two effective methods for solving high-dimensional partial
differential equations (PDE) based on randomized neural networks. Motivated by
the universal approximation property of this type of networks, both methods
extend the extreme learning machine (ELM) approach from low to high dimensions.
With the first method the unknown solution field in $d$ dimensions is
represented by a randomized feed-forward neural network, in which the
hidden-layer parameters are randomly assigned and fixed while the output-layer
parameters are trained. The PDE and the boundary/initial conditions, as well as
the continuity conditions (for the local variant of the method), are enforced
on a set of random interior/boundary collocation points. The resultant linear
or nonlinear algebraic system, through its least squares solution, provides the
trained values for the network parameters. With the second method the
high-dimensional PDE problem is reformulated through a constrained expression
based on an Approximate variant of the Theory of Functional Connections
(A-TFC), which avoids the exponential growth in the number of terms of TFC as
the dimension increases. The free field function in the A-TFC constrained
expression is represented by a randomized neural network and is trained by a
procedure analogous to the first method. We present ample numerical simulations
for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to
demonstrate their performance. These methods can produce accurate solutions to
high-dimensional PDEs, in particular with their errors reaching levels not far
from the machine accuracy for relatively lower dimensions. Compared with the
physics-informed neural network (PINN) method, the current method is both
cost-effective and more accurate for high-dimensional PDEs.

我们提出了两种基于随机神经网络解决高维偏微分方程 (PDE) 的有效方法。通过对这种类型网络的普适逼近性质的激励，这两种方法都将极限学习机 (ELM) 方法从低维扩展到高维。第一种方法中，$d$ 维度下未知解域由随机前向神经网络表示，其中隐藏层参数随机分配并固定，而输出层参数进行训练。PDE、边界 / 初值条件以及连续性条件 (对于方法的局部变量) 被施加在一组随机内部 / 边界对应点上。通过最小二乘解决其结果线性或非线性代数系统，从而得到网络参数的训练值。第二种方法通过一个基于近似理论的被约束表达式重新描述高维 PDE 问题，避免了随着维度增加而引发的 TFC 项数量的指数级增长。约束表达式中的自由域函数由随机神经网络表示，并通过类似于第一种方法的过程进行训练。我们进行了大量数值模拟，针对多个高维线性 / 非线性静态 / 动态 PDE，以展示这些方法的性能。与基于物理知识的神经网络 (PINN) 方法相比，当前方法在高维 PDEs 上既具有成本效益，又更准确。

一种基于极限学习机的高维计算 PDEs 方法

An Extreme Learning Machine-Based Method for Computational PDEs in  Higher Dimensions

Deep neural networks have been shown to provide accurate function
approximations in high dimensions. However, fitting network parameters requires
training data that may not be available beforehand, which is particularly
challenging in science and engineering applications where often it is even
unclear how to collect new informative training data in the first place. This
work proposes Neural Galerkin schemes based on deep learning that generate
training data samples with active learning for numerically solving
high-dimensional partial differential equations. Neural Galerkin schemes train
networks by minimizing the residual sequentially over time, which enables
adaptively collecting new training data in a self-informed manner that is
guided by the dynamics described by the partial differential equations, which
is in stark contrast to many other machine learning methods that aim to fit
network parameters globally in time without taking into account training data
acquisition. Our finding is that the active form of gathering training data of
the proposed Neural Galerkin schemes is key for numerically realizing the
expressive power of networks in high dimensions. Numerical experiments
demonstrate that Neural Galerkin schemes have the potential to enable
simulating phenomena and processes with many variables for which traditional
and other deep-learning-based solvers fail, especially when features of the
solutions evolve locally such as in high-dimensional wave propagation problems
and interacting particle systems described by Fokker-Planck and kinetic
equations.

该研究提出了一种新的基于深度学习的神经 Galerkin 方法，用于数值求解高维偏微分方程。该方法可以自适应地采集新的训练数据，以实现在高维空间中训练深度神经网络，从而成功地模拟了许多变量的系统中的波动和相互作用。

高维演化方程的神经 Galerkin 方案与主动学习

Neural Galerkin Scheme with Active Learning for High-Dimensional  Evolution Equations

High-dimensional partial differential equations (PDEs) are ubiquitous in
economics, science and engineering. However, their numerical treatment poses
formidable challenges since traditional grid-based methods tend to be
frustrated by the curse of dimensionality. In this paper, we argue that tensor
trains provide an appealing approximation framework for parabolic PDEs: the
combination of reformulations in terms of backward stochastic differential
equations and regression-type methods in the tensor format holds the promise of
leveraging latent low-rank structures enabling both compression and efficient
computation. Following this paradigm, we develop novel iterative schemes,
involving either explicit and fast or implicit and accurate updates. We
demonstrate in a number of examples that our methods achieve a favorable
trade-off between accuracy and computational efficiency in comparison with
state-of-the-art neural network based approaches.

本文研究高维偏微分方程，提出了一种基于张量列的近似方法和迭代方案，能够在精确性和计算效率之间达到有利的折衷。