Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solving Partial Differential Equations with Equivariant Extreme Learning Machines

Hans Harder, Jean Rabault, Ricardo Vinuesa, Mikael Mortensen, Sebastian Peitz

TL;DR

The paper tackles data-efficient PDE modeling by introducing a sliding-window, non-recurrent surrogate based on extreme-learning machines (ELMs) to learn the flow map of chaotic PDEs. A single two-layer ELM is trained with random first-layer features and a linear output layer, optionally augmented with positional encodings to handle spatial inhomogeneity, and is applied uniformly across spatial windows. It extends to two-dimensional Cahn–Hilliard and Kuramoto–Sivashinsky equations, demonstrating that symmetry-based equivariance—via the Euclidean group and its square-subgroup—can significantly reduce data needs and boost prediction quality. The approach offers a lightweight, differentiable surrogate that delivers long-horizon predictions from minimal data, aided by a spectral solver implementation and open-source code.

Abstract

We utilize extreme-learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data points (in some cases, our method can learn from a single full-state snapshot), it still achieves high accuracy and can predict the flow of PDEs over long time horizons. Moreover, we show how additional symmetries can be exploited to increase sample efficiency and to enforce equivariance.

Solving Partial Differential Equations with Equivariant Extreme Learning Machines

TL;DR

The paper tackles data-efficient PDE modeling by introducing a sliding-window, non-recurrent surrogate based on extreme-learning machines (ELMs) to learn the flow map of chaotic PDEs. A single two-layer ELM is trained with random first-layer features and a linear output layer, optionally augmented with positional encodings to handle spatial inhomogeneity, and is applied uniformly across spatial windows. It extends to two-dimensional Cahn–Hilliard and Kuramoto–Sivashinsky equations, demonstrating that symmetry-based equivariance—via the Euclidean group and its square-subgroup—can significantly reduce data needs and boost prediction quality. The approach offers a lightweight, differentiable surrogate that delivers long-horizon predictions from minimal data, aided by a spectral solver implementation and open-source code.

Abstract

We utilize extreme-learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data points (in some cases, our method can learn from a single full-state snapshot), it still achieves high accuracy and can predict the flow of PDEs over long time horizons. Moreover, we show how additional symmetries can be exploited to increase sample efficiency and to enforce equivariance.
Paper Structure (11 sections, 3 theorems, 28 equations, 8 figures, 1 table)

This paper contains 11 sections, 3 theorems, 28 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extreme learning machines for PDEs
  4. Model architecture and training
  5. Experiments
  6. Model statistics (\ref{['eq:KS_1d']}).
  7. Using positional encodings (\ref{['eq:KS_1d_spatial']}).
  8. Learning from a single sample (\ref{['eq:CH_Nd', 'eq:KS_Nd']}).
  9. Exploiting more symmetries and enforcing equivariance
  10. Acknowledgement.
  11. Appendix

Key Result

lemma thmcounterlemma

If $v : \mathbb R^n \rightarrow \mathbb R$ is $m$-times differentiable and $\bm i \in J^m$, then

Figures (8)

  • Figure 1: Sliding-window approach with an extent of $\ell = 1$ and a stride of $s = 3$.
  • Figure 2: Exemplary simulation of the homogeneous KS system compared to an ELM's prediction (left) and statistics over multiple ($\sim 50$) models (right).
  • Figure 3: Simulation of the inhomogeneous KS system compared to the ELM's prediction with and without positional encoding (PE).
  • Figure 4: Statistics from the inhomogeneous KS system. The $k$-th raw moment of a trajectory $\overline v$ (simulated or predicted by one of the ELMs) is $\mathbb E[\overline v(\bm t)^k]$, where $\bm t$ is sampled uniformly from the time interval. Similar flow statistics were used by srinivasan2019predictions.
  • Figure 5: Simulation of the two-dimensional Cahn-Hilliard equation compared to the ELM's prediction (left), together with the training data (right).
  • ...and 3 more figures

Theorems & Definitions (6)

  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof