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Invariant Physics-Informed Neural Networks for Ordinary Differential Equations

Shivam Arora, Alex Bihlo, Francis Valiquette

TL;DR

This paper introduces invariant physics-informed neural networks for ordinary differential equations that admit a finite-dimensional group of Lie point symmetries and illustrates the method with several examples, all of which considerably outperform standard non-invariants.

Abstract

Physics-informed neural networks have emerged as a prominent new method for solving differential equations. While conceptually straightforward, they often suffer training difficulties that lead to relatively large discretization errors or the failure to obtain correct solutions. In this paper we introduce invariant physics-informed neural networks for ordinary differential equations that admit a finite-dimensional group of Lie point symmetries. Using the method of equivariant moving frames, a differential equation is invariantized to obtain a, generally, simpler equation in the space of differential invariants. A solution to the invariantized equation is then mapped back to a solution of the original differential equation by solving the reconstruction equations for the left moving frame. The invariantized differential equation together with the reconstruction equations are solved using a physics-informed neural network, and form what we call an invariant physics-informed neural network. We illustrate the method with several examples, all of which considerably outperform standard non-invariant physics-informed neural networks.

Invariant Physics-Informed Neural Networks for Ordinary Differential Equations

TL;DR

This paper introduces invariant physics-informed neural networks for ordinary differential equations that admit a finite-dimensional group of Lie point symmetries and illustrates the method with several examples, all of which considerably outperform standard non-invariants.

Abstract

Physics-informed neural networks have emerged as a prominent new method for solving differential equations. While conceptually straightforward, they often suffer training difficulties that lead to relatively large discretization errors or the failure to obtain correct solutions. In this paper we introduce invariant physics-informed neural networks for ordinary differential equations that admit a finite-dimensional group of Lie point symmetries. Using the method of equivariant moving frames, a differential equation is invariantized to obtain a, generally, simpler equation in the space of differential invariants. A solution to the invariantized equation is then mapped back to a solution of the original differential equation by solving the reconstruction equations for the left moving frame. The invariantized differential equation together with the reconstruction equations are solved using a physics-informed neural network, and form what we call an invariant physics-informed neural network. We illustrate the method with several examples, all of which considerably outperform standard non-invariant physics-informed neural networks.
Paper Structure (11 sections, 3 theorems, 93 equations, 6 figures, 1 table)

This paper contains 11 sections, 3 theorems, 93 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Previous work
  3. Method
  4. Invariant differential equations
  5. Invariantization
  6. Recurrence relations
  7. Reconstruction
  8. Summary
  9. Invariant physics-informed neural networks
  10. Examples
  11. Summary and conclusions

Key Result

Proposition 3

A nondegenerate ordinary differential equation $\Delta(z^{(n)})=0$ is strongly invariant under the prolonged action of a connected local Lie group of transformations $G$ if and only if where $\mathbf{v}_1,\ldots,\mathbf{v}_r$ is a basis of infinitesimal generators for the group of transformations $G$.

Figures (6)

  • Figure 1: Solving a differential equation using moving frames.
  • Figure 2: Time series of the squared error for the Schwarz equation \ref{['Schwarz']}.
  • Figure 3: Time series of the squared error for the logistic equation \ref{['logistic']}.
  • Figure 4: Time series of the squared error for the driven harmonic oscillator \ref{['driven oscillator']}.
  • Figure 5: Time series of the squared error for the exponential equation \ref{['eq:exponential']}.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Definition 1
  • Remark 2
  • Proposition 3
  • Remark 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Theorem 9
  • Remark 10
  • ...and 12 more