Tensor Neural Networks for Efficient Approximation of High-Dimensional Steady-State Fokker-Planck Equations

Core Concepts

Tensor neural networks, including tensor radial basis function networks and tensor feedforward networks, can efficiently solve high-dimensional steady-state Fokker-Planck equations by leveraging the tensor product structure to exploit auto-differentiation and numerical integration.

Abstract

The paper presents a methodology for solving high-dimensional steady-state Fokker-Planck equations using tensor neural networks. Key highlights:

Tensor neural networks, including tensor radial basis function networks (TRBFN) and tensor feedforward networks (TFFN), are used to approximate the solutions. The tensor product structure allows efficient exploitation of auto-differentiation and numerical integration.

A procedure is designed to estimate efficient numerical supports for the Fokker-Planck equations, which is crucial for high-dimensional problems.

For TRBFN, constraints are imposed on the parameters of the radial basis functions to improve the approximation accuracy.

The tensor neural networks are trained using physics-informed loss functions and stochastic gradient descent methods.

The proposed approach is demonstrated on several examples in 2 to 10 dimensions, showing the effectiveness of tensor neural networks in capturing the complex interactions and multi-modal nature of the Fokker-Planck solutions.

Comparisons between TRBFN and TFFN are provided, highlighting the advantages and trade-offs of each approach in high-dimensional settings.

Stats

The paper does not contain explicit numerical data or statistics. The focus is on the methodology and numerical experiments.

Quotes

"Tensor neural networks, including tensor radial basis function networks and tensor feedforward networks, can efficiently solve high-dimensional steady-state Fokker-Planck equations by leveraging the tensor product structure to exploit auto-differentiation and numerical integration."

Key Insights Distilled From

Tensor neural networks for high-dimensional Fokker-Planck equations

by Taorui Wang,... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.05615.pdf

Deeper Inquiries

How can the proposed tensor neural network approach be extended to time-dependent Fokker-Planck equations

The proposed tensor neural network approach can be extended to time-dependent Fokker-Planck equations by incorporating the time variable into the neural network architecture. In the context of time-dependent equations, the neural network would need to evolve over time to capture the dynamics of the system. This can be achieved by introducing a time-dependent component in the neural network structure, allowing it to learn the time evolution of the probability density function. By training the network on time-series data and adjusting the weights and parameters over time, the tensor neural network can effectively model the time-dependent behavior of the Fokker-Planck equation.

What are the theoretical guarantees on the approximation capabilities of tensor neural networks for high-dimensional Fokker-Planck equations

The theoretical guarantees on the approximation capabilities of tensor neural networks for high-dimensional Fokker-Planck equations can be analyzed based on the universal approximation properties of neural networks. Tensor neural networks, which are a tensor product of one-dimensional feedforward networks or a linear combination of radial basis functions, have been shown to be universal approximators. This means that they have the ability to approximate any continuous function on a compact set with arbitrary accuracy. Therefore, in the context of high-dimensional Fokker-Planck equations, tensor neural networks can provide accurate approximations of the probability density function in the high-dimensional space. The approximation capabilities can be further enhanced by adjusting the network architecture, training parameters, and numerical support to optimize the performance of the neural network in solving the Fokker-Planck equation.

Can the tensor neural network framework be applied to other types of high-dimensional partial differential equations beyond Fokker-Planck equations

The tensor neural network framework can be applied to other types of high-dimensional partial differential equations beyond Fokker-Planck equations. The flexibility and adaptability of tensor neural networks make them suitable for a wide range of high-dimensional PDEs in various fields such as physics, engineering, finance, and biology. By adjusting the network architecture, incorporating domain-specific knowledge, and optimizing the training process, tensor neural networks can effectively solve complex high-dimensional PDEs. Applications may include diffusion equations, wave equations, heat equations, and other types of PDEs that arise in different scientific and engineering disciplines. The key lies in tailoring the network design and training methodology to the specific characteristics and requirements of the given PDE problem.