# Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions

arXiv: https://arxiv.org/abs/2007.05384 · HTML: https://mlsanity.com/paper/2007.05384/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jul 2020.

## Built on by (most cited first, 25 of 25)

- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems](https://mlsanity.com/paper/2112.03749/) (2021)
- [Parametric Complexity Bounds for Approximating PDEs with Neural Networks](https://mlsanity.com/paper/2103.02138/) (2021)
- [Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective](https://mlsanity.com/paper/2210.12101/) (2022)
- [Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning](https://mlsanity.com/paper/2206.10588/) (2022)
- [Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent](https://mlsanity.com/paper/2205.07331/) (2022)
- [Multilevel CNNs for Parametric PDEs](https://mlsanity.com/paper/2304.00388/) (2023)
- [Neural Q-learning for solving PDEs](https://mlsanity.com/paper/2203.17128/) (2022)
- [Quantitative approximation results for complex-valued neural networks](https://mlsanity.com/paper/2102.13092/) (2021)
- [Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations](https://mlsanity.com/paper/2103.05744/) (2021)
- [Residual Multi-Fidelity Neural Network Computing](https://mlsanity.com/paper/2310.03572/) (2023)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Theoretical guarantees for neural control variates in MCMC](https://mlsanity.com/paper/2304.01111/) (2023)
- [From Monte Carlo to neural networks approximations of boundary value problems](https://mlsanity.com/paper/2209.01432/) (2022)
- [Approximation properties of Residual Neural Networks for Kolmogorov PDEs](https://mlsanity.com/paper/2111.00215/) (2021)
- [Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions](https://mlsanity.com/paper/2206.01255/) (2022)
- [From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMs](https://mlsanity.com/paper/2503.09986/) (2025)
- [Sampling Complexity of Deep Approximation Spaces](https://mlsanity.com/paper/2312.13379/) (2023)
- [FinNet: Solving Time-Independent Differential Equations with Finite Difference Neural Network](https://mlsanity.com/paper/2202.09282/) (2022)
- [A numerical approach for the fractional Laplacian via deep neural networks](https://mlsanity.com/paper/2308.16272/) (2023)
- [Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias](https://mlsanity.com/paper/2406.09194/) (2024)
- [Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Method](https://mlsanity.com/paper/2607.06369/) (2026)
- [Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing](https://mlsanity.com/paper/2608.09494/) (2026)
- [Residual neural networks overcome the curse of dimensionality for semilinear heat equations](https://mlsanity.com/paper/2609.03626/) (2026)

## Abstract (arXiv)

In recent work it has been established that deep neural networks are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain Dsubset ℝd subject to Dirichlet boundary conditions. It is shown that deep neural networks are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.