Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

Abstract

Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs. Theorem 1.1. Let c, L P r0, 8q, λ P pL, 8q, M P NXpp ? λ`?Lq 2 p ? λ´?Lq ´2, 8q, Θ " Y nPN Z n , let u d P C 2 pR d , Rq, d P N, and f d P CpR d ˆR, Rq, d P N, satisfy for all d P N, x P R d that p∆u d qpxq " f d px, u d pxqq, (1) let pΩ, F , Pq be a probability space, let W d,θ : r0, 8q ˆΩ Ñ R d , θ P Θ, d P N, be i.i.d. standard Brownian motions, let R θ : Ω Ñ r0, 8q, θ P Θ, be i.i.d. random variables, assume that pR θ q θPΘ and pW d,θ q pd,θqPNˆΘ are independent, assume for all d P N, x " px 1 , . . . , x d q P R d , v, w P R, ε P p0, 8q that |f d px, vq ´fd px, wq ´λpv ´wq| ď L|v ´w|, |f d px, 0q| ď cd c r1 řd j"1 |x j |s c , sup y"py 1 ,...,y d qPR d r|u d pyq| expp´ε ř d j"1 |y j |qs ă 8, and PpR 0 ě εq " e ´λε , let U d,θ n " pU d,θ n pxqq xPR d : R d ˆΩ Ñ R, θ P Θ, d, n P N 0 , satisfy for all d, n P N, θ P Θ, x P R d that U d,θ 0 pxq " 0 and U d,θ n pxq " ´1 λM n « M n ÿ m"1 f d `x `?2 W d,pθ,0,mq R pθ,0,mq , 0 ˘ff `n´1 ÿ k"1 1 λM pn´kq « M pn´kq ÿ m"1 ˆλ" U d,pθ,k,mq k `x `?2 W d,pθ,k,mq R pθ,k,mq ˘´U d,pθ,k,´mq k´1 `x `?2 W d,pθ,k,mq R pθ,k,mq ˘ı ´"f d ´x `?2 W d,pθ,k,mq R pθ,k,mq , U d,pθ,k,mq k `x `?2 W d,pθ,k,mq R pθ,k,mq ˘f d ´x `?2 W d,pθ,k,mq R pθ,k,mq , U d,pθ,k,´mq k´1 `x `?2 W d,pθ,k,mq R pθ,k,mq

Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space

“…) specifies a part of the running and g ∈ C 1 (R d ; R) the terminal costs, and (X u s ) t≤s≤T denotes the unique strong solution to the controlled SDE (5) with initial condition X u t = x init . Throughout we assume that f and g are such that the expectation in ( 7) is finite, for all (x init , t) ∈ R d × [0, T ].…”

“…Consequently, we can elevate divergences between path measures to loss functions on vector fields. To wit, let D : P(C) × P(C) → R ≥0 ∪ {+∞} be a divergence 5 , where, as before, P(C) denotes the set of probability measures on C. Then, setting…”

“…In recent years, approaches to approximating the solutions of high-dimensional elliptic and parabolic PDEs have been developed combining well-known Feynman-Kac formulae with machine learning methodologies, seeking scalability and robustness in high-dimensional and complex scenarios [50,111]. Crucially, the use of artificial neural networks offers the promise of accurate and efficient function approximation which in conjunction with Monte Carlo methods can beat the curse of dimensionality, as investigated in [5,25,49,60]. In this paper, we focus on HJB-PDEs that can be linked to controlled diffusions (see Section 2),…”

Space-time deep neural network approximations for high-dimensional partial differential equations

“…It is one of the most challenging issues in applied mathematics to approximately solve highdimensional partial differential equations (PDEs) and most of the numerical approximation methods for PDEs in the scientific literature suffer from the so-called curse of dimensionality in the sense that the number of computational operations employed in the corresponding approximation scheme to obtain an approximation precision ε > 0 grows exponentially in the PDE dimension and/or the reciprocal of ε (cf., e.g., [42, Chapter 1] and [43,Chapter 9] for related concepts and cf., e.g., [4,5,7,19,29,32,33] for numerical approximation methods for nonlinear PDEs which do not suffer from the curse of dimensionality). Recently, certain deep learning based approximation methods for PDEs have been proposed and various numerical simulations for such methods suggest (cf., e.g., [1,2,3,8,9,10,12,13,14,15,17,18,21,26,27,28,30,34,39,40,41,44,45,46,48]) that deep neural network (DNN) approximations might have the capacity to indeed overcome the curse of dimensionality in the sense that the number of real parameters used to describe the approximating DNNs grows at most polynomially in both the PDE dimension d ∈ N = {1, 2, .…”

“…Our proofs of Theorem 1.1 above and Theorem 4.13 below, respectively, are based on an application of Proposition 3.10 in Grohs et al [23] (see (I)-(VI) in the proof of Proposition 4.8 in Subsection 4. 4…”