Partial differential equations (PDEs) are a fundamental tool in the modeling of many real-world phenomena. In a number of such real-world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are high-dimensional. Such high-dimensional nonlinear PDEs can in nearly all cases not be solved explicitly, and it is one of the most challenging tasks in applied mathematics to solve high-dimensional nonlinear PDEs approximately. It is especially very challenging to design approximation algorithms for nonlinear PDEs for which one can rigorously prove that they do overcome the so-called curse of dimensionality in the sense that the number of computational operations of the approximation algorithm needed to achieve an approximation precision of size \({\varepsilon }> 0\) grows at most polynomially in both the PDE dimension \(d \in \mathbb {N}\) and the reciprocal of the prescribed approximation accuracy \({\varepsilon }\). In particular, to the best of our knowledge there exists no approximation algorithm in the scientific literature which has been proven to overcome the curse of dimensionality in the case of a class of nonlinear PDEs with general time horizons and gradient-dependent nonlinearities. It is the key contribution of this article to overcome this difficulty. More specifically, it is the key contribution of this article (i) to propose a new full-history recursive multilevel Picard approximation algorithm for high-dimensional nonlinear heat equations with general time horizons and gradient-dependent nonlinearities and (ii) to rigorously prove that this full-history recursive multilevel Picard approximation algorithm does indeed overcome the curse of dimensionality in the case of such nonlinear heat equations with gradient-dependent nonlinearities.

偏微分方程 (PDE) 是许多现实世界现象建模的基本工具。在许多这样的现实世界现象中，所考虑的偏微分方程包含依赖于梯度的非线性并且是高维的。这种高维非线性偏微分方程几乎在所有情况下都不能显式求解，近似求解高维非线性偏微分方程是应用数学中最具挑战性的任务之一。设计非线性偏微分方程的近似算法尤其具有挑战性，因为我们可以严格证明它们确实克服了所谓的维数灾难，因为近似算法的计算操作数量需要达到大小的近似精度。\({\varepsilon }> 0\)在 PDE 维度\(d \in \mathbb {N}\)和规定的近似精度的倒数\({\varepsilon }\)中最多以多项式增长. 特别是，据我们所知，科学文献中不存在已被证明可以克服一类具有一般时间范围和梯度相关非线性的非线性偏微分方程的维数灾难的近似算法。克服这个困难是本文的关键贡献。更具体地说，本文的主要贡献是 (i) 为具有一般时间范围和梯度相关非线性的高维非线性热方程提出了一种新的全历史递归多级 Picard 近似算法，以及 (ii) 严格证明在这种具有梯度相关非线性的非线性热方程的情况下，这种全历史递归多级皮卡德近似算法确实克服了维数灾难。