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Deep neural network approximation for high-dimensional parabolic partial integro-differential equations

Marcin Baranek

TL;DR

The paper proves the existence of a ReLU neural network that can approximate the solution to a high-dimensional parabolic integro-differential equation with a Lévy-integral term, by leveraging the Feynman-Kac representation that expresses the solution as expectations of path functionals of an SDE with jumps. It develops a discretization-based construction of the stochastic dynamics using Euler-type schemes with Poisson-jump structure, and then shows how the two fundamental expectations—$\mathbb{E}[g(X_T^{(t,x)})]$ and $\mathbb{E}[\int_t^T b(s,X_s^{(t,x)}) ds]$—can be approximated by Monte Carlo estimators represented as DNNs. The main theoretical result (Theorem th:main_theorem) provides an existence claim for a ReLU network $\phi$ that achieves $L^2$ error at most $\delta$ with explicit dimension- and error-dependent size bounds, under growth and regularity assumptions encoded in classes $\mathcal{B}$, $\mathcal{G}$, and $\mathcal{C}$. The work is purely theoretical, establishing a constructive framework to bypass the curse of dimensionality in high-dimensional integro-differential equations, though no numerical experiments are provided.

Abstract

In this article, we investigate the existence of a deep neural network (DNN) capable of approximating solutions to partial integro-differential equations while circumventing the curse of dimensionality. Using the Feynman-Kac theorem, we express the solution in terms of stochastic differential equations (SDEs). Based on several properties of classical estimators, we establish the existence of a DNN that satisfies the necessary assumptions. The results are theoretical and don't have any numerical experiments yet.

Deep neural network approximation for high-dimensional parabolic partial integro-differential equations

TL;DR

The paper proves the existence of a ReLU neural network that can approximate the solution to a high-dimensional parabolic integro-differential equation with a Lévy-integral term, by leveraging the Feynman-Kac representation that expresses the solution as expectations of path functionals of an SDE with jumps. It develops a discretization-based construction of the stochastic dynamics using Euler-type schemes with Poisson-jump structure, and then shows how the two fundamental expectations— and —can be approximated by Monte Carlo estimators represented as DNNs. The main theoretical result (Theorem th:main_theorem) provides an existence claim for a ReLU network that achieves error at most with explicit dimension- and error-dependent size bounds, under growth and regularity assumptions encoded in classes , , and . The work is purely theoretical, establishing a constructive framework to bypass the curse of dimensionality in high-dimensional integro-differential equations, though no numerical experiments are provided.

Abstract

In this article, we investigate the existence of a deep neural network (DNN) capable of approximating solutions to partial integro-differential equations while circumventing the curse of dimensionality. Using the Feynman-Kac theorem, we express the solution in terms of stochastic differential equations (SDEs). Based on several properties of classical estimators, we establish the existence of a DNN that satisfies the necessary assumptions. The results are theoretical and don't have any numerical experiments yet.
Paper Structure (16 sections, 12 theorems, 132 equations)

This paper contains 16 sections, 12 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximation algorithm
  4. Basics of the algorithm
  5. Proof
  6. Proof
  7. Approximation by deep neural networks
  8. Proof
  9. Proof
  10. Proof of the main result
  11. Appendix
  12. Proof of lemma \ref{['bound_for_x_norm']}
  13. Proof of lemma \ref{['bound_for_x_diff_norm']}
  14. Proof of lemma \ref{['sde.schema.error']}
  15. Proof of lemma \ref{['integral_schema']}
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\phi_1$ and $\phi_2$ be two ReLU DNNs with input dimensions $N_0^i\in\mathbb{N}$ and output dimensions $N_L^i\in\mathbb{N}$, $i=1,2$ such as $N_0^1=N_L^2$. There exists a ReLU DNN $\phi$ such that $\mathrm{size}\left( \phi\right) = 2\mathrm{size}\left( \phi_1\right) + 2\mathrm{size}\left( \p

Theorems & Definitions (14)

  • Definition 1: dnn_appro_theory
  • Lemma 1: dnn_appro_theory
  • Lemma 2: dnn_appro_theory
  • Theorem 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 4 more