Space-time error estimates for deep neural network approximations for differential equations
Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philipp Zimmermann
Abstract
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 as computational advertisement. In addition, it has recently been proposed to approximate solutions of high-dimensional partial differential equations (PDEs) by means of stochastic learning problems involving deep ANNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for ANN approximations for PDEs but do not provide error estimates for the entire space-time error for the considered ANN approximations. It is the subject of the main result of this article to provide space-time error estimates for deep ANN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain ANN calculus and (ii) on ANN approximation results for products of the form $[0,T]\times \mathbb {R}^{d}\ni (t,x){\kern -.5pt}\mapsto {\kern -.5pt} tx{\kern -.5pt}\in {\kern -.5pt} \mathbb {R}^{d}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>]</mml:mo> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>ℝ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo>∋</mml:mo> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>↦</mml:mo> <mml:mi>t</mml:mi> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>ℝ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:math> where $T{\kern -.5pt}\in {\kern -.5pt} (0,\infty )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:math> , $d{\kern -.5pt}\in {\kern -.5pt} \mathbb {N}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℕ</mml:mi> </mml:math> , which we both develop within this article.