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arxiv: 2512.15997 · v1 · pith:5HQLPX6Nnew · submitted 2025-12-17 · 💻 cs.LG

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

Robert Stephany , William Michael Anderson , Youngsoo Choi This is my paper

Pith reviewed 2026-05-21 16:16 UTC · model grok-4.3

classification 💻 cs.LG
keywords reduced order modelingLaSDIfinite difference schemerollout lossBurgers equationPDE approximationlong-term prediction
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The pith

A high-order finite-difference scheme paired with rollout loss keeps reduced-order models accurate over long time horizons on PDEs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops higher-order LaSDI to fix the rapid loss of accuracy that reduced-order models show when forecasting partial differential equations far ahead in time. It does this by adding a flexible high-order finite-difference approximation that uses multiple time derivatives and by introducing a rollout loss that trains the model on predictions spanning arbitrary lengths. The authors apply the method to the two-dimensional Burgers equation and show improved long-term behavior. A sympathetic reader would care because many physical simulations in fluids, climate, or engineering need reliable forecasts over extended periods without the cost of full-order solvers.

Core claim

By introducing a flexible high-order yet inexpensive finite-difference scheme and a rollout loss that trains ROMs to make accurate predictions over arbitrary time horizons, higher-order LaSDI maintains predictive power on parameterized families of PDEs such as the 2D Burgers equation.

What carries the argument

The high-order finite-difference scheme that incorporates multiple time derivatives together with the rollout loss for training over extended sequences.

If this is right

  • Reduced-order models trained this way sustain accuracy across longer simulation intervals without rapid error accumulation.
  • The same scheme and loss apply to other parameterized PDEs beyond the Burgers equation.
  • Training remains inexpensive while extending the reliable prediction window.
  • The approach supports families of solutions rather than single instances.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Real-time control or design optimization loops could use these models for extended forecasts where full simulations are too slow.
  • The method may transfer to other latent dynamical systems in machine learning that suffer from long-term drift.
  • Testing on higher-dimensional or chaotic PDEs would clarify where the higher-order terms stop helping.

Load-bearing premise

The high-order finite-difference scheme combined with rollout training produces meaningfully better long-horizon accuracy than existing LaSDI or other ROM methods on the 2D Burgers equation.

What would settle it

A direct comparison of long-horizon prediction errors on the 2D Burgers equation where the new method shows no clear improvement over standard LaSDI baselines would falsify the central claim.

Figures

Figures reproduced from arXiv: 2512.15997 by Robert Stephany, William Michael Anderson, Youngsoo Choi.

Figure 1
Figure 1. Figure 1: A schematic for a parameterized ROM. A ROM can be used to predict the future FOM state by a) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A schematic of Rollout. From left to right, we begin by encoding a discretization [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A schematic of Rollout. From left to right, we begin by encoding a discretization [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A schematic of HLaSDI’s architecture, consisting of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Relative error of (left) displacement and (right) velocity applying HLaSDI to the 1D Burger’s [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Solutions of FOM (solid lines) and Higher-order LaSDI (dashed lines) for the Burger’s equation [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Relative error of (left) displacement and (right) velocity applying HLaSDI to the Wave equation [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Solutions of FOM and Higher-order LaSDI for the wave equation [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Standard deviation for our predictions of (left) [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Relative error of (left) displacement and (right) velocity HLaSDI to the Telegrapher’s equation [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Solutions of FOM and Higher-order LaSDI for the Telegrapher’s equation [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Relative error of (left) displacement and (right) velocity applying HLaSDI to the Klein-Gordon [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Solutions of FOM and Higher-order LaSDI for the Klein-Gordon equation [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The relative error of the displacement reconstruction when applying HLaSDI to the 2D Burgers [PITH_FULL_IMAGE:figures/full_fig_p029_14.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces Higher-Order LaSDI, extending Latent Space Dynamics Identification by incorporating multiple time derivatives through a flexible high-order finite-difference scheme and a rollout loss to train reduced-order models for accurate predictions over arbitrary time horizons. The approach is demonstrated on the 2D Burgers equation.

Significance. If the high-order scheme and rollout loss demonstrably improve long-horizon accuracy on nonlinear PDEs without introducing instability or excessive cost, the work would meaningfully advance ROM applicability in scientific computing for long-time simulations.

major comments (3)
  1. [Abstract] Abstract: The demonstration on the 2D Burgers equation is described without quantitative error metrics, baselines, implementation details, or comparisons to standard LaSDI, making it impossible to evaluate whether the proposed methods support the central claim of improved long-horizon predictions.
  2. [Finite-difference scheme description] Section on the finite-difference scheme: The claim that the high-order FD scheme is both accurate for multiple derivatives and inexpensive is load-bearing for the contribution, yet higher-order stencils increase stencil width and are known to amplify noise or require finer sampling; the manuscript provides no analysis of sensitivity to snapshot noise or discretization error in the 2D Burgers data.
  3. [Rollout loss formulation] Section on the rollout loss: The rollout loss is presented as addressing long-horizon degradation, but the manuscript does not show how it mitigates errors introduced by the high-order derivative estimation; without this, the combined method's robustness for arbitrary horizons remains unverified.
minor comments (2)
  1. [Notation and preliminaries] Clarify notation for the multiple time derivatives and how the FD scheme is embedded in the LaSDI framework.
  2. [Numerical results] Add a table or figure comparing computational cost and accuracy against baseline LaSDI variants.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript. We have carefully considered each major comment and will make revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The demonstration on the 2D Burgers equation is described without quantitative error metrics, baselines, implementation details, or comparisons to standard LaSDI, making it impossible to evaluate whether the proposed methods support the central claim of improved long-horizon predictions.

    Authors: We agree that the abstract would benefit from more quantitative details to support our claims. In the revised manuscript, we will update the abstract to include specific quantitative error metrics for the 2D Burgers equation experiments, such as time-averaged relative errors, direct comparisons to standard LaSDI, and brief notes on the implementation details and baselines. This will allow readers to better assess the improvements in long-horizon predictions. revision: yes

  2. Referee: [Finite-difference scheme description] Section on the finite-difference scheme: The claim that the high-order FD scheme is both accurate for multiple derivatives and inexpensive is load-bearing for the contribution, yet higher-order stencils increase stencil width and are known to amplify noise or require finer sampling; the manuscript provides no analysis of sensitivity to snapshot noise or discretization error in the 2D Burgers data.

    Authors: The referee correctly identifies a potential limitation. Although our high-order scheme is formulated to be flexible and we chose orders that balance accuracy and cost for the smooth 2D Burgers solutions, we did not provide an explicit sensitivity analysis to noise or discretization errors. We will add this analysis in the revised manuscript, including a discussion of how the scheme performs under varying levels of snapshot noise and grid resolutions, to substantiate the claims of accuracy and computational inexpensiveness. revision: yes

  3. Referee: [Rollout loss formulation] Section on the rollout loss: The rollout loss is presented as addressing long-horizon degradation, but the manuscript does not show how it mitigates errors introduced by the high-order derivative estimation; without this, the combined method's robustness for arbitrary horizons remains unverified.

    Authors: We appreciate this point regarding the interplay between the rollout loss and derivative estimation. The rollout loss is intended to train the dynamics model to be robust over multiple time steps, thereby reducing the impact of any inaccuracies in the estimated derivatives. To make this clearer, we will revise the relevant section to include a more detailed explanation and additional results showing the effect of the rollout loss on mitigating derivative estimation errors, for example by ablating the rollout component and comparing long-term prediction accuracy. revision: yes

Circularity Check

0 steps flagged

No circularity: new FD scheme and rollout loss are independent proposals

full rationale

The paper introduces a high-order finite-difference scheme for multiple time derivatives and a Rollout loss for long-horizon ROM training as methodological contributions, then demonstrates them on the 2D Burgers equation. These elements do not reduce to self-definitional fits, renamed inputs, or load-bearing self-citations by construction. The derivation chain presents the schemes as novel and inexpensive additions to LaSDI-style ROMs without equating any prediction or result to its own fitted parameters or prior author work in a circular manner. The central claims remain independent of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text.

pith-pipeline@v0.9.0 · 5616 in / 1101 out tokens · 48365 ms · 2026-05-21T16:16:04.640070+00:00 · methodology

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