发表机构
ETH Zurich; LMU Munich（苏黎世联邦理工学院; 慕尼黑大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出切尔诺夫-神经算子、包络-神经算子，通过迭代传播单步近似误差实现对强连续凸单调半群的通用近似，推导了定量近似率，并在多类数值例子中验证了方法的有效性。
AI 中文摘要
我们通过学习强连续凸单调半群的切尔诺夫型单步算子，用神经算子对其进行近似。首先，我们引入一类通用的所谓切尔诺夫-神经算子，并通过通用近似定理证明它们可以任意好地近似切尔诺夫单步算子。利用加权赫尔德空间之间的稳定性估计，单步近似误差可在迭代中传播，从而实现对应半群的通用近似。其次，针对包络半群，我们引入更专门的包络-神经算子类，这使我们能够推导定量近似率。最后，我们通过来自非线性偏微分方程、随机最优控制和模型不确定性下随机过程的多个数值例子，说明这些神经算子的有效性。
英文摘要
We approximate strongly continuous convex monotone semigroups by learning their Chernoff-type one-step operators with neural operators. First, we introduce the general class of so-called Chernoff-neural operators and show in a universal approximation theorem that they can approximate the Chernoff one-step operators arbitrarily well. By using stability estimates between weighted Hölder spaces, the one-step approximation error can be propagated through the iterations which yields universal approximation of the corresponding semigroup. Second, we introduce the more specialized class of envelope-neural operators for envelope semigroups which allows us to derive quantitative approximation rates. Finally, we illustrate the effectiveness of these neural operators in several numerical examples arising from non-linear partial differential equations, stochastic optimal control and stochastic processes under model uncertainty.
Comments38 pages, 6 figures
