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

arXiv: https://arxiv.org/abs/2512.25017 · HTML: https://mlsanity.com/paper/2512.25017/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 39 references that are in the map.
- First posted on arXiv 31 Dec 2025.

## Builds on (most cited first, 20 of 20)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
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- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Analysis of Stochastic Gradient Descent in Continuous Time](https://mlsanity.com/paper/2004.07177/) (2020)
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- [Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem](https://mlsanity.com/paper/1710.04273/) (2017)
- [Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.06000/) (2023)
- [A Continuous-time Stochastic Gradient Descent Method for Continuous Data](https://mlsanity.com/paper/2112.03754/) (2021)
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- [A time-stepping deep gradient flow method for option pricing in (rough) diffusion models](https://mlsanity.com/paper/2403.00746/) (2024)
- [Losing momentum in continuous-time stochastic optimisation](https://mlsanity.com/paper/2209.03705/) (2022)
- [A new approach to generalisation error of machine learning algorithms: Estimates and convergence](https://mlsanity.com/paper/2306.13784/) (2023)
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## Abstract (arXiv)

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.