残差神经网络克服半线性热方程的维数灾难
Residual neural networks overcome the curse of dimensionality for semilinear heat equations
- University of Freiburg（弗赖堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明残差神经网络可克服半线性热方程的维数灾难，给出其参数数量的显式界，通过ResNet实现多级皮卡估计器的确定性完成近似任务。
AI中文摘要：
严格结果表明，前馈神经网络可在高维偏微分方程（PDE）的数值近似中克服维数灾难，但残差神经网络（ResNets）在非线性PDE场景中的相关研究相对较少。本文证明，ResNets可在具有全局Lipschitz连续、与梯度无关的非线性项的半线性热方程的数值近似中克服维数灾难：在PDE数据的多项式增长和网络可近似性假设下，存在η∈(0,∞)，以及ResNets Ψ_{d,ε}（d∈ℕ，ε∈(0,1]），其参数数量最多为η d^η ε^{-η}，可在维度d下以L²误差不超过ε近似解。该证明将多级皮卡估计器的一个确定性实现表示为ResNet，其跳跃连接传递空间变量和标量累加器，而残差分支依次添加估计器的求和项。对于脊和初始条件、可容许的S型激活函数以及非线性项的全局Lipschitz截断，我们对每个ξ>0，得到参数数量的显式界C_ξ d^{4+ξ} ε^{-(3+ξ)}。
英文摘要：
Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting. We prove that ResNets overcome the curse of dimensionality in the numerical approximation of solutions of semilinear heat equations with globally Lipschitz continuous, gradient-independent nonlinearities: under polynomial growth and network approximability hypotheses on the PDE data, there exist $η\in(0,\infty)$ and ResNets $Ψ_{d,\varepsilon}$, $d\in\mathbb{N}$, $\varepsilon\in(0,1]$, with at most $ηd^η\varepsilon^{-η}$ parameters whose realizations approximate the solution in dimension $d$ with an $L^2$-error of at most $\varepsilon$. The proof represents one deterministic realization of a multilevel Picard estimator by a ResNet whose shortcut connections transmit the spatial variable and a scalar accumulator, while the residual branches successively add the summands of the estimator. For ridge-sum initial conditions, admissible sigmoidal activations, and globally Lipschitz truncations of the nonlinearity, we obtain, for every $ξ>0$, the explicit bound $C_ξd^{4+ξ}\varepsilon^{-(3+ξ)}$ on the number of parameters.
