Neural Parametric Fokker-Planck Equations

Liu, Shu; Li, Wuchen; Zha, Hongyuan; Zhou, Haomin

doi:10.48550/arxiv.2002.11309

Cited by 5 publications

(9 citation statements)

References 33 publications

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“…These approaches have been widely applied to many realistic problems, such as fluid mechanics [29,2], high dimensional PDEs (with applications in computational finance) [11,41], uncertainty quantification [39,27,42,14,22], to name just a few. Meanwhile, generative models such as generative adversarial networks [9], variational autoencoder [17] and normalizing flow (NF) [24,30], have also been successfully applied to learn forward and inverse PDEs [3,43,40,20]. For instance, physics-informed generative adversarial model was proposed in [38] to tackle high dimensional stochastic differential equations.…”

Section: Introductionmentioning

confidence: 99%

Solving time dependent Fokker-Planck equations via temporal normalizing flow

Xing¹,

Zeng²,

Zhou³

2021

Preprint

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In this work, we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck (TFP) equations. It is well known that solutions of such equations are probability density functions, and thus our approach relies on modelling the target solutions with the temporal normalizing flows. The temporal normalizing flow is then trained based on the TFP loss function, without requiring any labeled data. Being a machine learning scheme, the proposed approach is mesh-free and can be easily applied to high dimensional problems. We present a variety of test problems to show the effectiveness of the learning approach.

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Section: Introductionmentioning

confidence: 99%

Solving time dependent Fokker-Planck equations via temporal normalizing flow

Xing¹,

Zeng²,

Zhou³

2021

Preprint

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“…While traditional gridbased numerical methods, such as finite element methods and finite difference methods [61,32,56] can be employed to solve the Fokker-Plank equation, they are usually limited to low-dimensional problems. On the other hand, neural network-based methods has been successfully used in solving high dimensional PDEs [29,18,36,69,52,30,70,17], including the recent application in solving the high-dimensional Fokker-Plank equation [66,70,35]. These successes encourage us to also use deep learning to solve the approximate Fokker-Planck PDE.…”

Section: Introductionmentioning

confidence: 99%

Stationary Density Estimation of Itô Diffusions Using Deep Learning

Harlim

Liang

et al. 2021

Preprint

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In this paper, we consider the density estimation problem associated with the stationary measure of ergodic Itô diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take an advantage of the characterization of density function through the stationary solution of a parabolic-type Fokker-Planck PDE, we proceed as follows. First, we employ deep neural networks to approximate the drift and diffusion terms of the SDE by solving appropriate supervised learning tasks. Subsequently, we solve a steady-state Fokker-Plank equation associated with the estimated drift and diffusion coefficients with a neural-network-based least-squares method. We establish the convergence of the proposed scheme under appropriate mathematical assumptions, accounting for the generalization errors induced by regressing the drift and diffusion coefficients, and the PDE solvers. This theoretical study relies on a recent perturbation theory of Markov chain result that shows a linear dependence of the density estimation to the error in estimating the drift term, and generalization error results of nonparametric regression and of PDE regression solution obtained with neural-network models. The effectiveness of this method is reflected by numerical simulations of a two-dimensional Student's t distribution and a 20-dimensional Langevin dynamics. K eywordsStochastic differential equations • Data-driven method • Deep neural network • Fokker-Plank equation

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“…Natural gradient descent has been proven to be advantageous in various problems in machine learning and statistical inference, such as blind source separation [3], reinforcement learning [30] and neural network training [25,24,23,34,29,15,35,19]. Further applications include solution methods for high-dimensional Fokker-Planck equations [16,21]. It is a natural framework to include the curvature of the loss function based on the natural Riemannian structure of the ρ-space [2], and thus accelerate the rate of convergence in the θ-space and directly control the model in the ρ-space instead of the θ-space.…”

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confidence: 99%

“…These methods take advantage of the computational aspects of feed-forward neural networks by recycling computations during the forward and backward passes through the network. Furthermore, computational methods for natural gradients generated by the Wasserstein metric in the ρ-space are either variational [15,7,19,16,21,41] or relying on the entropic regularization of the Wasserstein distance [35]. The variational techniques are based on the proximal approximation (1.2), whereas entropic regularization techniques utilize computationally efficient Sinkhorn divergences [35].…”

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confidence: 99%

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Efficient Natural Gradient Descent Methods for Large-Scale Optimization Problems

Nurbekyan¹,

Lei²,

Yang³

2022

Preprint

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We propose an efficient numerical method for computing natural gradient descent directions with respect to a generic metric in the state space. Our technique relies on representing the natural gradient direction as a solution to a standard least-squares problem. Hence, instead of calculating, storing, or inverting the information matrix directly, we apply efficient methods from numerical linear algebra to solve this least-squares problem. We treat both scenarios where the derivative of the state variable with respect to the parameter is either explicitly known or implicitly given through constraints. We apply the QR decomposition to solve the least-squares problem in the former case and utilize the adjoint-state method to compute the natural gradient descent direction in the latter case. As a result, we can reliably compute several natural gradient descents, including the Wasserstein natural gradient, for a large-scale parameter space with thousands of dimensions, which was believed to be out of reach. Finally, our numerical results shed light on the qualitative differences among the standard gradient descent method and various natural gradient descent methods based on different metric spaces in large-scale nonconvex optimization problems.

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Neural Parametric Fokker-Planck Equations

Cited by 5 publications

References 33 publications

Solving time dependent Fokker-Planck equations via temporal normalizing flow

Solving time dependent Fokker-Planck equations via temporal normalizing flow

Stationary Density Estimation of Itô Diffusions Using Deep Learning

Efficient Natural Gradient Descent Methods for Large-Scale Optimization Problems

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