Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives

TL;DR

This paper introduces Higher-Order LaSDI, a reduced-order modeling technique that employs a high-order finite-difference scheme and a Rollout loss to improve long-term predictive accuracy for PDEs, demonstrated on the 2D Burgers equation.

Contribution

It presents a novel high-order finite-difference scheme combined with a Rollout loss for training ROMs, enhancing long-term prediction capabilities.

Findings

Improved long-term accuracy in PDE predictions.

Effective on the 2D Burgers equation.

Reduced computational cost compared to traditional methods.

Abstract

Solving complex partial differential equations is vital in the physical sciences, but often requires computationally expensive numerical methods. Reduced-order models (ROMs) address this by exploiting dimensionality reduction to create fast approximations. While modern ROMs can solve parameterized families of PDEs, their predictive power degrades over long time horizons. We address this by (1) introducing a flexible, high-order, yet inexpensive finite-difference scheme and (2) proposing a Rollout loss that trains ROMs to make accurate predictions over arbitrary time horizons. We demonstrate our approach on the 2D Burgers equation.