Convergence of the generalization error for deep gradient flow methods for PDEs

ArXiv ID: 2512.25017 “View on arXiv”

Authors: Chenguang Liu, Antonis Papapantoleon, Jasper Rou

Abstract

The aim of this article is to provide a firm mathematical foundation for the application of deep gradient flow methods (DGFMs) for the solution of (high-dimensional) partial differential equations (PDEs). We decompose the generalization error of DGFMs into an approximation and a training error. We first show that the solution of PDEs that satisfy reasonable and verifiable assumptions can be approximated by neural networks, thus the approximation error tends to zero as the number of neurons tends to infinity. Then, we derive the gradient flow that the training process follows in the ``wide network limit’’ and analyze the limit of this flow as the training time tends to infinity. These results combined show that the generalization error of DGFMs tends to zero as the number of neurons and the training time tend to infinity.

Keywords: Partial Differential Equations (PDEs), Deep Gradient Flow, Generalization Error, Neural Networks, High-Dimensional Problems, Quantitative Finance

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats
• Why: The paper is dominated by advanced mathematical analysis involving Sobolev spaces, operator theory, and limits of gradient flows, which is characteristic of high-complexity theory. It lacks any empirical results, datasets, backtests, or implementation details, focusing purely on mathematical convergence proofs without practical validation.

  flowchart TD
    A["Research Goal<br>Establish theoretical foundation for<br>Deep Gradient Flow Methods (DGFMs)<br>for high-dimensional PDEs"] --> B["Methodology: Error Decomposition"]
    B --> C["Approximation Error Analysis<br>Neural Network approximation<br>convergence as neurons → ∞"]
    B --> D["Training Error Analysis<br>Derive Gradient Flow in<br>Wide Network Limit"]
    C --> E["Key Assumptions<br>Verifiable PDE properties<br>reasonable solution regularity"]
    D --> F["Limit Analysis<br>Flow convergence as<br>training time → ∞"]
    E --> G["Computational Process<br>Unified Analysis: Generalization Error"]
    F --> G
    G --> H["Key Findings<br>Generalization error → 0<br>as neurons → ∞ AND time → ∞<br><br>Validates DGFMs for<br>high-dimensional PDEs<br>(e.g., Quantitative Finance)"]