mlsanityData to 7 Oct 2026

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

arXiv 2002.02600 ↗First posted 7 Feb 2020cs.LG cs.NA math.NAcited by 35

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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.

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

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Citing papers per year: 2021: 6; 2022: 3; 2023: 7; 2024: 5; 2025: 9; 2026: 5.

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  1. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  2. 2022Finite Expression Method for Solving High-Dimensional Partial Differential Equationscited by 21
  3. 2021Reproducing Activation Function for Deep Learningcited by 19
  4. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  5. 2021A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equationscited by 18
  6. 2025Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approachcited by 18
  7. 2024Operator SVD with Neural Networks via Nested Low-Rank Approximationcited by 12
  8. 2022Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descentcited by 11
  9. 2022Neural Networks Based on Power Method and Inverse Power Method for Solving Linear Eigenvalue Problemscited by 7
  10. 2024Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problemscited by 7
  11. 2023A Policy Gradient Framework for Stochastic Optimal Control Problems with Global Convergence Guaranteecited by 6
  12. 2023The Galerkin method beats Graph-Based Approaches for Spectral Algorithmscited by 6
23 more
  1. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
  2. 2024Neural networks for bifurcation and linear stability analysis of steady states in partial differential equationscited by 5
  3. 2025A brief review of the Deep BSDE method for solving high-dimensional partial differential equationscited by 5
  4. 2021A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networkscited by 4
  5. 2023Computing excited states of molecules using normalizing flowscited by 4
  6. 2024Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Methodcited by 3
  7. 2021Recurrent Neural Networks for Stochastic Control Problems with Delaycited by 2
  8. 2023On the uncertainty analysis of the data-enabled physics-informed neural network for solving neutron diffusion eigenvalue problemcited by 2
  9. 2023Matrix Diagonalization as a Board Game: Teaching an Eigensolver the Fastest Path to Solutioncited by 2
  10. 2023A Deep-Genetic Algorithm (Deep-GA) Approach for High-Dimensional Nonlinear Parabolic Partial Differential Equationscited by 2
  11. 2025From Equations to Insights: Unraveling Symbolic Structures in PDEs with LLMscited by 2
  12. 2025Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equationcited by 2
  13. 2025STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problemcited by 2
  14. 2025Rethinking Neural-based Matrix Inversion: Why can't, and Where cancited by 1
  15. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0
  16. 2025NeuMatC: A General Neural Framework for Fast Parametric Matrix Operationcited by 0
  17. 2025Learning Eigenstructures of Unstructured Data Manifoldscited by 0
  18. 2025Deep Eigenspace Network for Parametric Non-self-adjoint Eigenvalue Problemscited by 0
  19. 2026Deep Neural networks for solving high-dimensional parabolic partial differential equationscited by 0
  20. 2026Beyond Bellman: High-Order Generator Regression for Continuous-Time Policy Evaluationcited by 0
  21. 2026Conflict-Aware Harmonized Rotational Gradient for Multiscale Kinetic Regimescited by 0
  22. 2026A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equationscited by 0
  23. 2026Machine learning kinetics from molecular dynamics datacited by 0