# On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs

arXiv: https://arxiv.org/abs/2004.01806 · HTML: https://mlsanity.com/paper/2004.01806/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Apr 2020.

## Built on by (most cited first, 30 of 31)

- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Physics informed neural networks for continuum micromechanics](https://mlsanity.com/paper/2110.07374/) (2021)
- [Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.16371/) (2025)
- [Physics-informed neural networks for the shallow-water equations on the sphere](https://mlsanity.com/paper/2104.00615/) (2021)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations](https://mlsanity.com/paper/2103.06183/) (2021)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Hybrid FEM-NN models: Combining artificial neural networks with the finite element method](https://mlsanity.com/paper/2101.00962/) (2021)
- [Learning Physics-Informed Neural Networks without Stacked Back-propagation](https://mlsanity.com/paper/2202.09340/) (2022)
- [Reproducing Activation Function for Deep Learning](https://mlsanity.com/paper/2101.04844/) (2021)
- [Solving time dependent Fokker-Planck equations via temporal normalizing flow](https://mlsanity.com/paper/2112.14012/) (2021)
- [State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning](https://mlsanity.com/paper/2202.06416/) (2022)
- [Neural oscillators for generalization of physics-informed machine learning](https://mlsanity.com/paper/2308.08989/) (2023)
- [Neural Q-learning for solving PDEs](https://mlsanity.com/paper/2203.17128/) (2022)
- [Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions](https://mlsanity.com/paper/2208.08635/) (2022)
- [ST-PINN: A Self-Training Physics-Informed Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2306.09389/) (2023)
- [Neural Control of Parametric Solutions for High-dimensional Evolution PDEs](https://mlsanity.com/paper/2302.00045/) (2023)
- [Convergence Analysis of the Deep Galerkin Method for Weak Solutions](https://mlsanity.com/paper/2302.02405/) (2023)
- [Physics Informed Convex Artificial Neural Networks (PICANNs) for Optimal Transport based Density Estimation](https://mlsanity.com/paper/2104.01194/) (2021)
- [Auxiliary-Tasks Learning for Physics-Informed Neural Network-Based Partial Differential Equations Solving](https://mlsanity.com/paper/2307.06167/) (2023)
- [Moving Sampling Physics-informed Neural Networks induced by Moving Mesh PDE](https://mlsanity.com/paper/2311.16167/) (2023)
- [Numerical PDE solvers outperform neural PDE solvers](https://mlsanity.com/paper/2507.21269/) (2025)
- [Accelerating Dynamical System Simulations with Contracting and Physics-Projected Neural-Newton Solvers](https://mlsanity.com/paper/2106.02543/) (2021)
- [Informed Learning by Wide Neural Networks: Convergence, Generalization and Sampling Complexity](https://mlsanity.com/paper/2207.00751/) (2022)
- [Approximation of Solution Operators for High-dimensional PDEs](https://mlsanity.com/paper/2401.10385/) (2024)
- [Label Propagation for Physics-Informed Neural Networks and Physics-Informed Gaussian Processes](https://mlsanity.com/paper/2404.05817/) (2024)

## Abstract (arXiv)

Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encounted in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs. As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in C0. Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes H1. Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.