Input gradient annealing neural network for solving low-temperature Fokker-Planck equations

We present a novel yet simple deep learning approach, called input gradient annealing neural network (IGANN), for solving stationary Fokker-Planck equations. Traditional methods, such as finite difference and finite elements, suffer from the curse of dimensionality. Neural network based algorithms a...

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Bibliographic Details
Published in:arXiv.org 2024-05
Main Authors: Liangkai Hang, Hu, Dan, Zin-Qin John Xu
Format: Article
Language:English
Subjects:
Algorithms
Annealing
Eigenvalues
Finite element method
Fokker-Planck equation
Low temperature
Machine learning
Mathematical analysis
Meshless methods
Metastable state
Neural networks
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Summary:We present a novel yet simple deep learning approach, called input gradient annealing neural network (IGANN), for solving stationary Fokker-Planck equations. Traditional methods, such as finite difference and finite elements, suffer from the curse of dimensionality. Neural network based algorithms are meshless methods, which can avoid the curse of dimensionality. However, at low temperature, when directly solving a stationary Fokker-Planck equation with more than two metastable states in the generalized potential landscape, the small eigenvalue introduces numerical difficulties due to a large condition number. To overcome these problems, we introduce the IGANN method, which uses a penalty of negative input gradient annealing during the training. We demonstrate that the IGANN method can effectively solve high-dimensional and low-temperature Fokker-Planck equations through our numerical experiments.
ISSN:2331-8422