# A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks

arXiv: https://arxiv.org/abs/2105.03480 · HTML: https://mlsanity.com/paper/2105.03480/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 22 references that are in the map.
- First posted on arXiv 7 May 2021.

## Builds on (most cited first, 10 of 10)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Speech Recognition with Deep Recurrent Neural Networks](https://mlsanity.com/paper/1303.5778/) (2013)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach](https://mlsanity.com/paper/2002.02600/) (2020)
- [Solving for high dimensional committor functions using artificial neural networks](https://mlsanity.com/paper/1802.10275/) (2018)
- [A semigroup method for high dimensional committor functions based on neural network](https://mlsanity.com/paper/2012.06727/) (2020)

## Built on by (most cited first, 4 of 4)

- [Operator SVD with Neural Networks via Nested Low-Rank Approximation](https://mlsanity.com/paper/2402.03655/) (2024)
- [Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method](https://mlsanity.com/paper/2410.00835/) (2024)
- [Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains](https://mlsanity.com/paper/2510.22497/) (2025)
- [Deep Eigenspace Network for Parametric Non-self-adjoint Eigenvalue Problems](https://mlsanity.com/paper/2512.20058/) (2025)

## Abstract (arXiv)

In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original equations as variational problems with the help of semigroup operators and then solve the variational problems with neural network (NN) parameterization. The main advantages are that no mixed second-order derivative computation is needed during the stochastic gradient descent training and that the boundary conditions are taken into account automatically by the semigroup operator. Unlike popular methods like PINN citeraissi2019physics and Deep Ritz citeweinan2018deep where the Dirichlet boundary condition is enforced solely through penalty functions and thus changes the true solution, the proposed method is able to address the boundary conditions without penalty functions and it gives the correct true solution even when penalty functions are added, thanks to the semigroup operator. For eigenvalue problems, a primal-dual method is proposed, efficiently resolving the constraint with a simple scalar dual variable and resulting in a faster algorithm compared with the BSDE solver citehan2020solving in certain problems such as the eigenvalue problem associated with the linear Schr"odinger operator. Numerical results are provided to demonstrate the performance of the proposed methods.