Space-time error estimates for deep neural network approximations for differential equations

Abstract Over the last few years deep artificial neural networks (ANNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well advertisement. In addition, it has recently proposed to approximate solutions high-dimensional partial differential equations (PDEs) by means stochastic learning involving ANNs. There are now also rigorous mathematical results scientific literature which provide error estimates such based approximation methods PDEs. All these articles spatial ANN approximations PDEs but do not entire space-time considered approximations. It is subject main result this article Euler certain perturbed equations. Our proof (i) on calculus and (ii) products form $[0,T]\times \mathbb {R}^{d}\ni (t,x){\kern -.5pt}\mapsto {\kern -.5pt} tx{\kern -.5pt}\in {R}^{d}$ [ 0 , T ] × ℝ d ∋ ( t x ) ↦ ∈ where $T{\kern (0,\infty )$ ∞ , $d{\kern {N}$ ℕ we both develop within article.