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Deep ReLU neural networks overcome the curse of dimensionality when approximating semilinear partial integro-differential equations

Ariel Neufeld, Tuan Anh Nguyen, Sizhou Wu

TL;DR

This work proves that deep ReLU neural networks can approximate the unique viscosity solution of high-dimensional semilinear PIDEs with jumps without suffering the curse of dimensionality. The authors couple stochastic fixed point representations with multilevel Picard (MLP) approximations and show these approximations can be realized as DNNs whose size scales polynomially in the dimension $d$ and the inverse accuracy $\epsilon^{-1}$. By constructing approximants for the linear, nonlinear, and nonlocal terms as DNNs and proving stability of the SFPE, they obtain a global neural network $\Psi_{d,\epsilon}$ achieving an $L^2$ error $\leq \epsilon$ with a polynomial parameter bound $\mathcal{P}(\Psi_{d,\epsilon})$. This provides a rigorous theoretical foundation for using DNN-based solvers to tackle high-dimensional nonlinear PIDEs encountered in finance and related fields. The approach unifies SFPE representations, MLP convergence, and DNN realizations to overcome dimensionality barriers in nonlinear, nonlocal PDEs.

Abstract

In this paper we consider PIDEs with gradient-independent Lipschitz continuous nonlinearities and prove that deep neural networks with ReLU activation function can approximate solutions of such semilinear PIDEs without curse of dimensionality in the sense that the required number of parameters in the deep neural networks increases at most polynomially in both the dimension $ d $ of the corresponding PIDE and the reciprocal of the prescribed accuracy $ε$.

Deep ReLU neural networks overcome the curse of dimensionality when approximating semilinear partial integro-differential equations

TL;DR

This work proves that deep ReLU neural networks can approximate the unique viscosity solution of high-dimensional semilinear PIDEs with jumps without suffering the curse of dimensionality. The authors couple stochastic fixed point representations with multilevel Picard (MLP) approximations and show these approximations can be realized as DNNs whose size scales polynomially in the dimension and the inverse accuracy . By constructing approximants for the linear, nonlinear, and nonlocal terms as DNNs and proving stability of the SFPE, they obtain a global neural network achieving an error with a polynomial parameter bound . This provides a rigorous theoretical foundation for using DNN-based solvers to tackle high-dimensional nonlinear PIDEs encountered in finance and related fields. The approach unifies SFPE representations, MLP convergence, and DNN realizations to overcome dimensionality barriers in nonlinear, nonlocal PDEs.

Abstract

In this paper we consider PIDEs with gradient-independent Lipschitz continuous nonlinearities and prove that deep neural networks with ReLU activation function can approximate solutions of such semilinear PIDEs without curse of dimensionality in the sense that the required number of parameters in the deep neural networks increases at most polynomially in both the dimension of the corresponding PIDE and the reciprocal of the prescribed accuracy .
Paper Structure (16 sections, 18 theorems, 175 equations)

This paper contains 16 sections, 18 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Existence and uniqueness of solutions of PIDEs
  4. Setting for approximating functions
  5. A mathematical framework for DNNs
  6. Main result
  7. Outline of the proof and organization of the paper
  8. Stochastic fixed point equations
  9. Approximation of the coefficients
  10. Euler-Maruyama and MLP approximations revisited
  11. DNNs
  12. Properties of operations associated to DNNs
  13. DNN representation of our Euler-Maruyama approximations
  14. DNN representation of our MLP approximations
  15. Proof of \ref{['k41']}
  16. ...and 1 more sections

Key Result

Proposition 1.2

Assume setting PIDE. Then for every $d\in {\mathbbm{N}}$ there exists a unique viscosity solutionFor the definition of a viscosity solution see, e.g., NW2022.$u^d\colon [0,T]\times {\mathbbm{R}}^d\to{\mathbbm{R}}$ to the PIDE satisfying that $\sup_{s\in [0,T],y\in {\mathbbm{R}}^d}\frac{\lvert u^d(s,y)\rvert}{(d^{c}+\lVert y\rVert^2)^{1/2}}<\infty$.

Theorems & Definitions (27)

  • Proposition 1.2
  • Theorem 1.5
  • Lemma 2.1
  • proof : Proof of \ref{['lemma f epsilon']}
  • Proposition 2.2: Stochastic fixed point equations
  • proof : Proof of \ref{['Prop SFPE']}
  • Proposition 3.1: A stability result
  • proof : Proof of \ref{['x01']}
  • Proposition 4.1: Discretization error
  • proof : Proof of \ref{['k32']}
  • ...and 17 more