  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.
