论文标题
PDE的神经网络近似超过线性：代表性的观点
Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective
论文作者
论文摘要
一条迅速发展的研究线利用深层神经网络将解决方案近似于高维PDE的解决方案，理论探究的开场白着重于解释这些模型如何逃避维度的诅咒。但是，大多数先前的理论分析仅限于线性PDE。在这项工作中，我们朝着研究神经网络的代表力迈出了一步，以近似于非线性PDE的解决方案。我们专注于一类称为\ emph {非线性椭圆形变异pdes}的PDE，其解决方案最小化\ emph {euler -lagrange}能量函数$ \ MATHCAL {e}（e}（u）= \int_Ωl（\int_Ωl）我们表明，如果用Barron Norm $ b $组成$ l $的部分衍生物的功能最多可以产生Barron Norm的函数，那么PDE的解决方案可以是$ε$ - 在$ l^2 $中通过Barron Norm norm norm norm norm n norm norm $ o \ weft（\ lest（db_ left） p^{\ log（1/ε）}}}} \ right）$。通过Barron [1993]引起的经典结果，这相应地界定了近似溶液所需的2层神经网络的大小。将$ p，ε，b_l $视为常数，该数量在维度上是多项式，因此显示神经网络可以逃避维度的诅咒。我们的证明技术涉及在适当的希尔伯特空间中的神经模拟（预处理）梯度，该梯度将指数迅速收敛到PDE的溶液，因此我们可以绑定每个迭代中Barron Norm的增加。我们的结果集中并在单位超数据集上概括了线性椭圆PDE的类似先验结果。
A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most prior theoretical analyses have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as \emph{nonlinear elliptic variational PDEs}, whose solutions minimize an \emph{Euler-Lagrange} energy functional $\mathcal{E}(u) = \int_ΩL(x, u(x), \nabla u(x)) - f(x) u(x)dx$. We show that if composing a function with Barron norm $b$ with partial derivatives of $L$ produces a function of Barron norm at most $B_L b^p$, the solution to the PDE can be $ε$-approximated in the $L^2$ sense by a function with Barron norm $O\left(\left(dB_L\right)^{\max\{p \log(1/ ε), p^{\log(1/ε)}\}}\right)$. By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating $p, ε, B_L$ as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.
