A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations

Hornung, Fabian; Jentzen, Arnulf; Wurstemberger, Philippe von

doi:10.48550/arxiv.1809.02362

Cited by 77 publications

(142 citation statements)

References 57 publications

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“…Remark.. • There is a common belief that Machine learning based PDE solvers can break the curse of dimensionality [15,24,40]. However we obtained an n − 2s−2 2s−4+d convergence rate which can become super slow in high dimension.…”

Section: Final Upper Boundmentioning

confidence: 76%

Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality

Lu¹,

Chen²,

Lu³

et al. 2021

Preprint

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In this paper, we study the statistical limits of deep learning techniques for solving elliptic partial differential equations (PDEs) from random samples using the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). To simplify the problem, we focus on a prototype elliptic PDE: the Schrödinger equation on a hypercube with zero Dirichlet boundary condition, which is applied in quantum-mechanical systems. We establish upper and lower bounds for both methods, which improve upon concurrently developed upper bounds for this problem via a fast rate generalization bound. We discover that the current Deep Ritz Method is sub-optimal and propose a modified version of it. We also prove that PINN and the modified version of DRM can achieve minimax optimal bounds over Sobolev spaces. Empirically, following recent work which has shown that the deep model accuracy will improve with growing training sets according to a power law, we supply computational experiments to show similar-behavior of dimension dependent power law for deep PDE solvers.

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Section: Final Upper Boundmentioning

confidence: 76%

Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality

Lu¹,

Chen²,

Lu³

et al. 2021

Preprint

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“…Since that time there appeared many other versions of the Kolmogorov-Chentsov theorem that essentially allow to treat more general sets Θ. We mention [14, Theorem 2.1], [5, Theorem 3.9], [7,Lemma 2.19], [9, Proposition 3.9] for several recent formulations, where Θ is a subset of R m . Sometimes only the existence of a continuous or Höldercontinuous modification is claimed, which is substantially weaker than (1.4).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A general Kolmogorov-Chentsov type theorem with applications to limit theorems for Banach-valued processes

Krätschmer¹,

Urusov²

2021

Preprint

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The paper deals with moduli of continuity for paths of random processes with outcomes in general metric spaces. Adapting the moment condition on the increments from the classical Kolmogorov-Chentsov theorem, the obtained result on the modulus of continuity allows for Hölder-continuous modifications if the metric spaces are complete. This result is universal in the sense that its applicability depends only on the geometry of the parametric spaces. In particular, it is always applicable if parametric spaces are bounded subsets of Euclidean spaces. The derivation is based on refined chaining techniques developed by Talagrand. As a consequence of the main result a criterion is presented to guarantee uniform tightness of random processes with continuous paths. This is applied to find central limit theorems for Banach-valued random processes.

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“…Several theoretical work have been devoted to the above representation question. It has been established in [14,15,20] that deep neural networks can approximate solutions to certain class of parabolic equations and Poisson equation without CoD. The major limitation of those work lies in that the PDEs considered in those work must admit certain stochastic representation such as the Feymann-Kac formula and it seems difficult to generalize the proof techniques to broader classes of PDEs with no probabilistic interpretation.…”

Section: Main Theorem (Informal Version)mentioning

confidence: 99%

On the Representation of Solutions to Elliptic PDEs in Barron Spaces

Chen¹,

Lu²,

Lu³

2021

Preprint

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Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of d-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is ǫ-close with respect to the H 1 norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension d of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the H 1 norm with a dimension-explicit convergence rate.

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A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations

Cited by 77 publications

References 57 publications

Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality

Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality

A general Kolmogorov-Chentsov type theorem with applications to limit theorems for Banach-valued processes

On the Representation of Solutions to Elliptic PDEs in Barron Spaces

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