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Tensor neural networks for high-dimensional Fokker-Planck equations

Taorui Wang, Zheyuan Hu, Kenji Kawaguchi, Zhongqiang Zhang, George Em Karniadakis

TL;DR

It is demonstrated numerically that the tensor neural networks in physics-informed machine learning are efficient for steady-state Fokker-Planck equations from two to ten dimensions.

Abstract

We solve high-dimensional steady-state Fokker-Planck equations on the whole space by applying tensor neural networks. The tensor networks are a linear combination of tensor products of one-dimensional feedforward networks or a linear combination of several selected radial basis functions. The use of tensor feedforward networks allows us to efficiently exploit auto-differentiation (in physical variables) in major Python packages while using radial basis functions can fully avoid auto-differentiation, which is rather expensive in high dimensions. We then use the physics-informed neural networks and stochastic gradient descent methods to learn the tensor networks. One essential step is to determine a proper bounded domain or numerical support for the Fokker-Planck equation. To better train the tensor radial basis function networks, we impose some constraints on parameters, which lead to relatively high accuracy. We demonstrate numerically that the tensor neural networks in physics-informed machine learning are efficient for steady-state Fokker-Planck equations from two to ten dimensions.

Tensor neural networks for high-dimensional Fokker-Planck equations

TL;DR

It is demonstrated numerically that the tensor neural networks in physics-informed machine learning are efficient for steady-state Fokker-Planck equations from two to ten dimensions.

Abstract

We solve high-dimensional steady-state Fokker-Planck equations on the whole space by applying tensor neural networks. The tensor networks are a linear combination of tensor products of one-dimensional feedforward networks or a linear combination of several selected radial basis functions. The use of tensor feedforward networks allows us to efficiently exploit auto-differentiation (in physical variables) in major Python packages while using radial basis functions can fully avoid auto-differentiation, which is rather expensive in high dimensions. We then use the physics-informed neural networks and stochastic gradient descent methods to learn the tensor networks. One essential step is to determine a proper bounded domain or numerical support for the Fokker-Planck equation. To better train the tensor radial basis function networks, we impose some constraints on parameters, which lead to relatively high accuracy. We demonstrate numerically that the tensor neural networks in physics-informed machine learning are efficient for steady-state Fokker-Planck equations from two to ten dimensions.
Paper Structure (31 sections, 1 theorem, 29 equations, 8 figures, 21 tables, 2 algorithms)

This paper contains 31 sections, 1 theorem, 29 equations, 8 figures, 21 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our proposed methodology and contributions
  3. Literature review
  4. General issues in high dimensions
  5. Solving Fokker-Planck equations with neural networks
  6. Problem and methodology
  7. Estimating Bounded Domain
  8. Tensor neural network (TNN)
  9. Tensor Feed-Forward Networks
  10. Tensor Radial Basis Function Networks
  11. Loss function
  12. Boundary Condition
  13. Training settings
  14. Training Platform
  15. Data Precision
  16. ...and 16 more sections

Key Result

Theorem B.1

Let $C_P$ be the constant from the Poincare inequality. Assume that where $C_D$ is from eq:uniform-ellipticity, $\mathbf{b}$ is defined above and $C_0$ is a positive constant. Then we have

Figures (8)

  • Figure 1: Flowchart of the proposed methodology for solving the Fokker-Planck equation
  • Figure 2: Example \ref{['exm:2d-ring']}: history of loss function. In TRBFN, we use the Wendland kernel.
  • Figure 3: Contour maps of projections of the exact solution $p^*(x)$ in Example \ref{['exm:4.2-4dunimode']}
  • Figure 4: Contour maps of projections of the exact solution in Example \ref{['exm:6d-multi-mode']}
  • Figure 5: Example \ref{['exm:6d-multi-mode']}: Loss history, TRBFN(800, 6).
  • ...and 3 more figures

Theorems & Definitions (9)

  • Remark 3.1
  • Remark 3.2
  • Example 4.1: 2D, ring potential
  • Example 4.2: 4D, solution with a single mode
  • Example 4.3: 6D, solution with a single mode
  • Example 4.4: 6D, solution with multiple modes
  • Example 4.5: 10D, solution with two modes
  • Theorem B.1
  • proof