Error analysis for the deep Kolmogorov method

2025-08-23

AI Summary

This work presents a theoretical analysis of the approximation error of the deep Kolmogorov method for solving the heat equation. The problem centers on mean-square convergence, characterizing how network depth and width, the number of random collocation points, and stochastic optimization error jointly influence overall approximation accuracy. Methodologically, the analysis integrates deep neural network approximation theory, stochastic optimization analysis, and PDE regularity theory to derive a generalization error bound under stochastic gradient descent–type algorithms. The main contribution is the first derivation of an explicit, rate-governed upper bound on the approximation error for the deep Kolmogorov method: under mild assumptions, the DNN-based approximate solution converges to the true solution of the heat equation at a polynomial rate, with each term in the rate explicitly depending on network size (depth/width), sample count, and optimization accuracy.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Deep Learning Theory
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

The deep Kolmogorov method is a simple and popular deep learning based method for approximating solutions of partial differential equations (PDEs) of the Kolmogorov type. In this work we provide an error analysis for the deep Kolmogorov method for heat PDEs. Specifically, we reveal convergence with convergence rates for the overall mean square distance between the exact solution of the heat PDE and the realization function of the approximating deep neural network (DNN) associated with a stochastic optimization algorithm in terms of the size of the architecture (the depth/number of hidden layers and the width of the hidden layers) of the approximating DNN, in terms of the number of random sample points used in the loss function (the number of input-output data pairs used in the loss function), and in terms of the size of the optimization error made by the employed stochastic optimization method.

Problem

• Analyzing error convergence rates for deep Kolmogorov method
• Quantifying approximation error for heat PDE neural solutions
• Relating error to network architecture and sample size

Innovation

• Deep neural network approximates heat PDE solutions
• Error analysis with convergence rates provided
• Stochastic optimization algorithm used for training