深度神经网络与多层皮卡德法可高效求解高维非线性偏微分方程。

Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense

• 采用多层皮卡德迭代与带ReLU等激活函数的深层网络
• 计算量与参数量随维度和精度要求呈多项式增长
• 突破高维问题计算复杂度瓶颈，适合高维金融/物理建模

我们证明，当非线性项为梯度无关且Lipschitz连续时，多层皮卡德逼近与使用ReLU、leaky ReLU及softplus激活函数的深度神经网络，能够以$ L^rak{p} $-范数（$ rak{p} imes [2,ty) $）逼近半线性柯尔莫哥洛夫型偏微分方程的解。在此情形下，多层皮卡德方法的计算代价以及神经网络所需参数数量，均在维度 $ d imes b{N} $ 和预设精度 $ rak{e} $ 的倒数之间至多呈多项式增长。

原文摘要 · Abstract (English)

We prove that multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation are capable of approximating solutions of semilinear Kolmogorov PDEs in $L^\mathfrak{p}$-sense, $\mathfrak{p}\in [2,\infty)$, in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the computational effort of the multilevel Picard approximations and the required number of parameters in the neural networks grow at most polynomially in both dimension $d\in \mathbb{N}$ and reciprocal of the prescribed accuracy $ε$.

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