Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

Abstract

For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy.

Three ways to solve partial differential equations with neural networks — A review

“…Beginning around 2016, the spectacular successes of machine learning systems in computer vision, natural language processing and other areas prompted renewed efforts to establish a mathematically rigorous foundation for, in particular, deep feedforward neural networks [14,22,49,61,62,75,99,118,123,128,134,142‐144,180,181]. We draw particular attention to a number of publications that rigorously establish that certain neural network architectures are theoretically able to overcome the curse of dimensionality for various linear and nonlinear PDEs, cf [8,18,65,81,82,84,85,92].…”

Overcoming the Curse of Dimensionality in the Numerical Approximation of Parabolic Partial Differential Equations with Gradient-Dependent Nonlinearities

“…Branching diffusion approximation methods are also in the case of certain nonlinear PDEs as efficient as plain vanilla Monte Carlo approximations in the case of linear PDEs, but the error analysis only applies in the case where the time horizon T ∈ (0, ∞) and the initial condition, respectively, are sufficiently small and branching diffusion approximation methods fail to converge in the case where the time horizon T ∈ (0, ∞) exceeds a certain threshold (cf., e.g., [41,Theorem 3.12]). For MLP approximation methods it has been recently shown in [4,45,46] that such algorithms do indeed overcome the curse of dimensionality for certain classes of gradient-independent PDEs. Numerical simulations for deep learning based approximation methods for nonlinear PDEs in high dimensions are very encouraging (see, e.g., the above named references [1][2][3][6][7][8]13,14,16,17,21,24,25,31,[34][35][36]40,43,48,50,52,[54][55][56][57][58]60,62,63]) but so far there is only partial error analysis available for such algorithms (which, in turn, is strongly based on the above-mentioned error analysis for the MLP approximation method; cf.…”

“…[44] and, e.g., [9,23,32,33,36,49,51,61,62]). To sum up, to the best of our knowledge until today the MLP approximation method (see [45]) is the only approximation method in the scientific literature for which it has been shown that it does overcome the curse of dimensionality in the numerical approximation of semilinear PDEs with general time horizons.…”

“…The above-mentioned articles [4,28,45,46] prove, however, only in the case of gradient-independent nonlinearities that MLP approximation methods overcome the curse of dimensionality and it remains an open problem to overcome the curse of dimensionality in the case of PDEs with general time horizons and gradient-dependent nonlinearities. This is precisely the subject of this article.…”

“…random variables satisfies for all b ∈ (0, 1) that P(r 0 ≤ b) = √ b and Corollary 5.1 and Theorem 5.2 are proved under the more general hypothesis that there exists α ∈ (0, 1) such that for all b ∈ (0, 1) it holds that P(r 0 ≤ b) = b α (see, e.g., (160) in Theorem 5.2). It is a crucial observation of this article that, roughly speaking, in the case of PDEs with gradientdependent nonlinearities time integrals in the semigroup formulations of the PDEs under consideration cannot be approximated with continuous uniformly distributed random variables (corresponding to the case α = 1) as it was done, e.g., in [4,45,46]; cf. Sect.…”