Core Concepts
This paper proposes a new adaptive deep learning method called temporal KRnet (tKRnet) to efficiently approximate the probability density functions (PDFs) of state variables in stochastic dynamical systems governed by the Liouville equation.
Abstract
The paper focuses on developing an efficient deep learning method to solve the Liouville equation, which models the evolution of PDFs in stochastic dynamical systems. The key contributions are:
- Extending the Knothe-Rosenblatt (KRnet) model to a time-dependent setting (tKRnet) to approximate the time-varying PDFs.
- Proposing an adaptive training procedure for tKRnet, where the collocation points for training are generated iteratively using the approximate PDF at each step. This helps the collocation points become more consistent with the solution PDF over iterations.
- Introducing a temporal decomposition technique to improve the long-time integration of the Liouville equation.
- Providing theoretical analysis to bound the Kullback-Leibler (KL) divergence between the exact solution and the tKRnet approximation.
The authors demonstrate the effectiveness of the proposed method through numerical examples involving stochastic dynamical systems like the double gyre flow, Kraichnan-Orszag problem, Duffing oscillator, and Lorenz-96 system.
Stats
The paper presents numerical results comparing the tKRnet solution with the reference solution computed using the method of characteristics. The relative error and KL divergence between the two solutions are reported.
Quotes
"The main idea of deep learning methods for solving PDEs is to reformulate a PDE problem as an optimization problem and train deep neural networks to approximate the solution by minimizing the corresponding loss function."
"The idea of the normalizing flows is to construct an invertible mapping from a given simple distribution to the unknown distribution, such that the PDF of the unknown distribution can be obtained by the change of variables."