Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives
This work addresses computational efficiency in physical sciences by enhancing ROMs for long-term PDE simulations, though it appears incremental as it builds on existing ROM methods.
The paper tackles the problem of reduced-order models (ROMs) for solving parameterized PDEs, which degrade in accuracy over long time horizons, by introducing a high-order finite-difference scheme and a Rollout loss to improve long-term predictions, demonstrating results on the 2D Burgers equation.
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.