# Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning

arXiv: https://arxiv.org/abs/2012.08023 · HTML: https://mlsanity.com/paper/2012.08023/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Dec 2020.

## Built on by (most cited first, 18 of 18)

- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Finite Expression Method for Solving High-Dimensional Partial Differential Equations](https://mlsanity.com/paper/2206.10121/) (2022)
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- [A Finite Expression Method for Solving High-Dimensional Committor Problems](https://mlsanity.com/paper/2306.12268/) (2023)
- [Bayesian neural networks for weak solution of PDEs with uncertainty quantification](https://mlsanity.com/paper/2101.04879/) (2021)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Convergence Analysis of the Deep Galerkin Method for Weak Solutions](https://mlsanity.com/paper/2302.02405/) (2023)
- [Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysis](https://mlsanity.com/paper/2409.20383/) (2024)
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- [EgPDE-Net: Building Continuous Neural Networks for Time Series Prediction with Exogenous Variables](https://mlsanity.com/paper/2208.01913/) (2022)
- [DPG loss functions for learning parameter-to-solution maps by neural networks](https://mlsanity.com/paper/2506.18773/) (2025)
- [Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equation](https://mlsanity.com/paper/2508.21571/) (2025)
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- [Astral: training physics-informed neural networks with error majorants](https://mlsanity.com/paper/2406.02645/) (2024)
- [Energy stable neural network for gradient flow equations](https://mlsanity.com/paper/2309.10002/) (2023)
- [On Continuity of Robust and Accurate Classifiers](https://mlsanity.com/paper/2309.17048/) (2023)
- [Error analysis for the deep Kolmogorov method](https://mlsanity.com/paper/2508.17167/) (2025)
- [Randomized Neural Networks for Integro-Differential Equations with Application to Neutron Transport](https://mlsanity.com/paper/2604.13830/) (2026)

## Abstract (arXiv)

This paper proposes Friedrichs learning as a novel deep learning methodology that can learn the weak solutions of PDEs via a minmax formulation, which transforms the PDE problem into a minimax optimization problem to identify weak solutions. The name "Friedrichs learning" is for highlighting the close relationship between our learning strategy and Friedrichs theory on symmetric systems of PDEs. The weak solution and the test function in the weak formulation are parameterized as deep neural networks in a mesh-free manner, which are alternately updated to approach the optimal solution networks approximating the weak solution and the optimal test function, respectively. Extensive numerical results indicate that our mesh-free method can provide reasonably good solutions to a wide range of PDEs defined on regular and irregular domains in various dimensions, where classical numerical methods such as finite difference methods and finite element methods may be tedious or difficult to be applied.