# Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach

arXiv: https://arxiv.org/abs/2002.02600 · HTML: https://mlsanity.com/paper/2002.02600/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Feb 2020.

## Built on by (most cited first, 30 of 35)

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- [A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equations](https://mlsanity.com/paper/2311.11558/) (2023)
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- [Rethinking Neural-based Matrix Inversion: Why can't, and Where can](https://mlsanity.com/paper/2506.00642/) (2025)
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## Abstract (arXiv)

We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the semigroup flow induced by the operator, whose solution can be represented by Feynman-Kac formula in terms of forward-backward stochastic differential equations. The method shares a similar spirit with diffusion Monte Carlo but augments a direct approximation to the eigenfunction through neural-network ansatz. The criterion of fixed point provides a natural loss function to search for parameters via optimization. Our approach is able to provide accurate eigenvalue and eigenfunction approximations in several numerical examples, including Fokker-Planck operator and the linear and nonlinear Schr"odinger operators in high dimensions.