SMORE: Stability-Promoting Mesh-Agnostic Model Reduction for Time-Dependent PDEs

Yangyuan Li; Weichao Li; Shaowu Pan

cs.LGSep 27, 2026

SMORE: Stability-Promoting Mesh-Agnostic Model Reduction for Time-Dependent PDEs

Authors: Yangyuan Li, Weichao Li, Shaowu Pan

Organizations: Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute 110 8th Street, Troy, NY 12180, USA

Abstract

High-fidelity simulations of time-dependent partial differential equations (PDEs) are computationally expensive, motivating data-driven reduced-order surrogates for many-query tasks such as uncertainty quantification, design optimization, data assimilation, and optimal control. However, existing surrogate models often exhibit poor temporal stability, which can lead to unstable rollouts and exploding gradients during backpropagation, especially in multistep long-horizon forecasting. To address this, we propose SMORE, a mesh-agnostic framework for model order reduction of time-dependent PDEs. Its latent dynamics are trained with Lyapunov-guided stability regularization, which promotes stable long-horizon rollouts. We provide theoretical guarantees under the stated structural assumptions. Beyond forecasting PDE evolution, the learned latent dynamics, which are interpretable and linear or linear-quadratic, could bring benefits for downstream tasks such as data assimilation and optimal control. Moreover, our framework is capable of predicting continuous PDE solution fields from sparse measurements of the initial condition. We evaluate SMORE on a range of problems, including wave propagation, the Navier-Stokes equations, and the shallow water equations. Our results show that it improves long-horizon rollout generalization and empirical robustness, and achieves competitive accuracy at comparable parameter budgets relative to competitive baselines including DINo, FNO, CNO, and Transolver.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of SMORE. (a) Overall SMORE workflow. An INR autodecoder fits compact latent states to observed field values. The latent dynamics advances these states in time, and the coordinate decoder reconstructs the corresponding solution fields. Training uses scheduled sampling, whereas testing rolls out from the inferred initial state. (b) SMORE-Koopman latent dynamics (left) and SMORE-LRLQ latent dynamics (right). (c) Lyapunov-guided stability regularization and coordinate decoding. Stability is promoted by augmenting the dynamics loss $L_{\alpha}$ with Lyapunov-guided regularizers (left). The eigenvalue term $L_{\text{reg,eig}}$ penalizes positive eigenvalues of the symmetric part $\mathbf{W}_{s}$ of the linear operator and is used by both SMORE-Koopman and SMORE-LRLQ, while the conservation term $L_{\text{reg,cons}}$ , which encourages the energy-preserving structure of the quadratic tensor $\mathbf{Q}$ , is used only by SMORE-LRLQ. Their combination yields the modified loss $L_{\text{dyn}}$ . Coordinate-based decoding then evaluates the reconstructed field at arbitrary spatial query locations (right).

Dataset	Prediction target	Grid	$\Delta t$	Seq. length	Train seq.	Test seq.
2D Wave	two-component wave field	$64^{2}$	$0.25$ s	20	512	32
2D Navier–Stokes	vorticity	$64^{2}$	$1$ s	20	512	32
3D Spherical Shallow Water	height, vorticity	$64\times 128$	$1$ h	20	64	16

Table 1: Summary of the PDE benchmark datasets. Grid denotes the spatial sampling resolution, and sequence length denotes the number of snapshots in each trajectory.

Model	Wave In-horizon	Wave Extrapolation	Navier–Stokes In-horizon	Navier–Stokes Extrapolation	Shallow Water In-horizon	Shallow Water Extrapolation
FNO [ 19 , 73 ]	2.89e-04	7.37e-02	1.61e-03	2.43e-03	1.79e-03	5.22e-03
CNO [ 20 ]	1.99e-02	1.53e-01	1.52e-03	3.87e-03	8.03e-04	2.86e-03
Transolver [ 25 ]	7.82e-04	1.67e-02	4.13e-01	6.22e-01	1.71e-03	6.05e-03
DINo [ 32 ]	1.13e-03	1.38e-02	1.34e-03	2.32e-03	4.90e-04	6.49e-04
SMORE-Koopman	6.74e-05	3.22e-04	5.37e-02	9.78e-02	2.22e-04	5.34e-04
SMORE-LRLQ	1.25e-03	8.65e-04	1.01e-03	2.00e-03	5.86e-04	7.67e-04

Table 2: In-horizon and extrapolation MSE for long-horizon rollout prediction under approximately matched parameter budgets. Values are averaged over three random seeds, with the corresponding standard deviations reported in Appendix Table 8 . For each run, the checkpoint with the lowest finite overall test MSE over the 19 predicted snapshots is selected; the initial snapshot is excluded. Best values in each metric column are bolded, and second-best values are underlined.

Figure 2: Field-level extrapolation comparison at snapshot 15 (prediction step $n=14$ ). Within each benchmark, all models are compared using the same random seed, the same test trajectory, and the same prediction step $n$ . Rows show the ground truth followed by model predictions; columns show the normalized Wave field $u_{x}$ , Navier–Stokes vorticity, and Spherical Shallow Water vorticity. These snapshots confirm that the low MSE in Table 2 reflects actual structure recovery. The predicted fields track the real spatial patterns and their evolution rather than only a low average error.

Figure 3: Training robustness of SMORE-LRLQ across per-component latent dimension $p$ . (a) Spherical Shallow Water: rows denote stability configurations and columns denote per-component latent dimensions. Hatched cells report the first logged epoch with NaN; “Not observed” indicates that none was observed during the available training window. (b) Navier–Stokes: best finite overall test MSE over the 19 predicted snapshots for different stability configurations. Legend annotations give the relative MSE change from $p=50$ to $p=100$ for each curve.

Dataset	FULL	FULL without $L_{\mathrm{reg}}$	FULL without $\mathrm{clip}$	FULL without $\mathrm{low\mbox{-}rank}$
Navier–Stokes	2.337e-3	3.475e-3 (+48.7%)	2.796e-3 (+19.7%)	5.191e-3 (+122.2%)
Shallow Water	6.299e-4	6.403e-4 (+1.7%)	6.342e-4 (+0.7%)	5.585e-3 † (+786.8%)

Table 3: Leave-one-out ablation of the FULL SMORE-LRLQ configuration (all mechanisms enabled) at per-component latent dimension

p=100

. Each entry is the total MSE over the 19 predicted snapshots (snapshots 2–20) under the 1000-step initial latent fitting during evaluation; the value in parentheses is the relative change from FULL, where positive means higher error. The columns remove the Lyapunov-guided stability regularization, gradient clipping, and the low-rank quadratic parameterization respectively. A dagger (

\dagger

) marks a run that diverged to non-finite values during training; its best pre-divergence checkpoint is used for evaluation.

Dataset (model)	$100\%$	$50\%$	$10\%$	$4\%$	$2\%$	$1\%$
Wave (SMORE-Koopman)	2.01e-04	2.05e-04	1.98e-04	2.11e-04	6.96e-04	4.06e-02
Navier–Stokes (SMORE-LRLQ)	1.53e-03	1.53e-03	1.60e-03	1.78e-03	3.77e-03	2.48e-02
Shallow Water (SMORE-Koopman)	3.86e-04	3.94e-04	4.55e-04	6.71e-04	1.32e-03	1.56e-03

Table 4: Sparse-initialization performance under selected initial-snapshot observation ratios. Each entry reports overall MSE over the 19 predicted snapshots, averaged across three random seeds. The fitted initial snapshot is excluded from the metric. Only the retained samples of the first snapshot are used to infer the initial latent state; the subsequent rollout reconstructs complete spatial fields without retraining. The complete visibility sweep with standard deviations is reported in Appendix Table 12 .

Panel of Figure 4 — Figure 4: Sparse initialization at $10\%$ visibility for (a) Wave, (b) Navier–Stokes, and (c) Spherical Shallow Water. In each panel, the top row shows the ground-truth field at snapshot indices $n=0,4,9,14,19$ (labeled $t$ in the panels). The bottom-left image shows the sparse initial observation, which is the only test-time input, while the remaining bottom images show full-field predictions at the corresponding later snapshots. Only the observed samples from the initial snapshot are used to infer $\widehat{\alpha}_{0}$ ; the decoder and latent dynamics remain frozen. Wave and Spherical Shallow Water use SMORE-Koopman, whereas Navier–Stokes uses SMORE-LRLQ.

Model	Spatial execution	Parameters	Maximum successful grid	Grid points	Relative points
Transolver [ 25 ]	Full grid	358,007	$64\times 64$	4,096	$1.00\times$
DINo [ 32 ]	Full grid	356,444	$163\times 163$	26,569	$6.49\times$
FNO [ 19 , 73 ]	Full grid	336,653	$271\times 271$	73,441	$17.93\times$
CNO [ 20 ]	Full grid	326,184	$307\times 307$	94,249	$23.01\times$
SMORE-Koopman	PM64	355,165	$1714\times 1714$	2,937,796	$717.24\times$
SMORE-LRLQ	PM64	353,961	$1714\times 1714$	2,937,796	$717.24\times$

Table 5: Maximum successful square grids on one A100 40 GB GPU under the two-epoch capacity protocol, together with model parameter counts on Navier–Stokes. Each maximum successful grid is one step below a CUDA OOM failure (i.e., the next larger square grid runs out of memory). PM64 denotes pointwise minibatches of 64 coordinates in all decoder evaluations, including training autodecoding, test-time initial latent fitting, and evaluation field reconstruction. Grid points is the number of spatial points at the maximum successful grid, and the last column reports this count relative to Transolver.

Metric	Full-grid decoding	PM64
Recorded peak allocated memory (MB)	18,613.9	314.6
Time for 5,000 epochs	1 h 8 min 21 s	4 h 4 min 32 s
In-horizon MSE ( $\times 10^{-3}$ )	$1.008\pm 0.025$	$1.048\pm 0.014$
Extrapolation MSE ( $\times 10^{-3}$ )	$2.000\pm 0.199$	$2.111\pm 0.262$
Overall MSE ( $\times 10^{-3}$ )	$1.530\pm 0.101$	$1.607\pm 0.132$

Table 6: Observed resource and accuracy changes for

64\times 64

Navier–Stokes SMORE-LRLQ. Memory is the logged training-epoch peak allocated memory on rank 0. Elapsed times are measured over complete 5,000-epoch runs and refer to one representative run per execution setting; these timings exclude the final 1,000-step test stage. MSEs are mean

\pm

sample standard deviation over three random seeds.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Scheduled sampling for latent-dynamics training. The upper-left panel shows target latent codes at supervised time stamps. The intermediate panels illustrate the progressive replacement of target codes by model-generated states during training. Intermediate target states are used as teacher-forcing anchors with probability $\varepsilon_{t}$ ; as this probability decays, training increasingly relies on free-running latent rollouts. The bottom panels schematically contrast a poorly matched rollout without intermediate anchors (left) with an improved rollout obtained using scheduled sampling (right).

Panel of Figure 12 — Algorithm 3 Forward propagation of the neural ODE with scheduled sampling

Baseline models
Dataset	Model	Architecture
wa	FNO	$(M,W,L)=(12,10,4)$
	CNO	$(L,R,RN,CM)=(3,4,6,6)$
	Transolver	$(L,H,S)=(5,33,64)$
ns	FNO	$(M,W,L)=(12,17,4)$
	CNO	$(L,R,RN,CM)=(3,1,8,10)$

Appendix

Table 7: Architecture hyper-parameters used in the final comparisons. The abbreviations wa , ns , and sw denote 2D Wave, 2D Navier–Stokes, and Spherical Shallow Water, and “–” marks a parameter that does not apply. For the baseline operators, FNO is specified by

(M,W,L)

, where

M

is the number of retained Fourier modes per dimension,

W

the channel width, and

L

the number of Fourier layers; CNO by

(L,R,RN,CM)

, where

L

is the number of up/down-sampling layers,

R

the number of residual blocks per level,

RN

the number of residual blocks in the bottleneck, and

CM

the channel multiplier; and Transolver by

(L,H,S)

, where

L

is the number of layers,

H

the hidden dimension, and

S

the number of physics-attention slices. For the latent models, in which MLP denotes the DINo dynamics,

p

is the per-component latent dimension and

r=cp

the total latent dimension,

\mathrm{hdyn}

is the hidden width of the MLP latent-dynamics network,

\mathrm{hdec}

is the hidden width of the INR (MFN) decoder,

\ell_{\mathrm{dec}}

is the number of hidden linear layers of the decoder, and

\mathrm{qr}

is the quadratic rank

\kappa

of the low-rank factorization used by SMORE-LRLQ.

Dataset	Model	Parameters	In-horizon loss	Extrapolation loss
Wave	FNO	117,356	$2.89e-04\pm 1.09e-04$	$7.37e-02\pm 1.41e-02$
Wave	CNO	111,996	$1.99e-02\pm 4.34e-03$	$1.53e-01\pm 1.47e-02$
Wave	Transolver	128,328	$7.82e-04\pm 3.90e-04$	$1.67e-02\pm 4.29e-03$
Wave	DINo	126,928	$1.13e-03\pm 9.98e-05$	$1.38e-02\pm 1.01e-03$
Wave	SMORE-Koopman	128,045	$6.74e-05\pm 3.63e-05$	$3.22e-04\pm 1.54e-04$
Wave	SMORE-LRLQ	127,265	$1.25e-03\pm 8.22e-07$	$8.65e-04\pm 3.09e-06$

Appendix

Table 8: Detailed in-horizon and extrapolation MSE on unseen test trajectories. Values are mean

\pm

sample standard deviation over three random seeds. DINo, SMORE-Koopman, and SMORE-LRLQ use results after 1,000 initial latent optimization steps.

Figure 6: Per-snapshot test MSE of SMORE-LRLQ on the 2D Wave benchmark, averaged over three random seeds (individual seeds shown as thin lines). The dashed vertical line separates the in-horizon window (snapshots 2–10) from the extrapolation window (snapshots 11–20). The error rises to a peak inside the in-horizon window and decreases over much of the extrapolation window, so the extrapolation-window average lies below the in-horizon average even though no actual improvement occurs at longer horizons.

Configuration	$p=50$	$p=70$	$p=100$
OFF	8111	200	100
$L_{\text{reg,cons}}$ only	Not observed	147	142
$L_{\mathrm{reg,eig}}$ only	8100	227	135
$L_{\mathrm{reg}}$ only	Not observed	150	146
clip only	7162	158	100
low-rank only	Not observed	Not observed	Not observed

Appendix

Table 9: First observed NaN epoch in the Spherical Shallow Water mechanism study. “Not observed” indicates that no NaN was recorded during the full 10,000-epoch training budget. Remaining finite over this interval is an empirical observation and does not establish formal stability.

Configuration	$p=10$	$p=30$	$p=50$	$p=70$	$p=100$
OFF	3.139e-2	4.432e-3	1.966e-3	3.226e-3	7.215e-3
$L_{\text{reg,cons}}$ only	3.137e-2	4.496e-3	2.921e-3	4.141e-3	6.708e-3
$L_{\mathrm{reg,eig}}$ only	3.139e-2	4.506e-3	1.959e-3	3.351e-3	5.102e-3
$L_{\mathrm{reg}}$ only	3.137e-2	4.536e-3	2.845e-3	4.097e-3	7.018e-3
clip only	3.087e-2	4.441e-3	1.939e-3	2.295e-3	4.532e-3
low-rank only	3.960e-2	5.335e-3	2.635e-3	2.887e-3	4.161e-3

Appendix

Table 10: SMORE-LRLQ stability ablation on 2D Navier–Stokes. Values are the total rollout MSE over the 19 predicted snapshots (snapshots 2–20) under the 1000-step initial latent fitting during evaluation. The configured training budget is 10,000 epochs. The low-rank quadratic rank is

\lfloor p/6\rfloor

Configuration	$p=50$	$p=70$	$p=100$
OFF	0.8614 †	2.360 †	4.069 †
$\lambda_{1}$ only	0.8574	2.757 †	3.397 †
$\lambda_{2}$ only	0.8603 †	2.699 †	3.427 †
$\lambda$ only	0.8464	2.953 †	3.240 †
clip only	0.8843 †	2.669 †	5.313 †
low-rank only	0.9120	0.7832	0.7305

Appendix

Table 11: SMORE-LRLQ stability-mechanism ablation on 3D Spherical Shallow Water. Values are the best finite overall MSE over the 19 predicted snapshots, reported in units of

10^{-3}

. A dagger (

\dagger

) indicates that the run subsequently produced NaN; the reported value is the best finite MSE obtained before divergence. The low-rank rank is 10. All training runs were continued for up to 10,000 epochs unless numerical divergence occurred.

Visible ratio	Wave (SMORE-Koopman)	Navier–Stokes (SMORE-LRLQ)	Shallow Water (SMORE-Koopman)
$100\%$	2.01e-04 $\pm$ 9.84e-05	1.53e-03 $\pm$ 1.01e-04	3.86e-04 $\pm$ 8.58e-06
$50\%$	2.05e-04 $\pm$ 1.04e-04	1.53e-03 $\pm$ 1.09e-04	3.94e-04 $\pm$ 8.03e-06
$25\%$	2.12e-04 $\pm$ 1.05e-04	1.56e-03 $\pm$ 1.06e-04	4.06e-04 $\pm$ 2.01e-06
$20\%$	2.12e-04 $\pm$ 1.09e-04	1.57e-03 $\pm$ 9.51e-05	4.17e-04 $\pm$ 2.38e-06
$15\%$	2.01e-04 $\pm$ 8.59e-05	1.59e-03 $\pm$ 1.03e-04	4.30e-04 $\pm$ 4.23e-06
$10\%$	1.98e-04 $\pm$ 8.39e-05	1.60e-03 $\pm$ 1.03e-04	4.55e-04 $\pm$ 3.97e-06

Appendix

Table 12: Complete sparse-initialization results using 1,000 initial latent optimization steps with frozen checkpoints. Overall test MSE excludes the fitted initial snapshot and is evaluated on the full spatial grid. Values are mean

\pm

sample standard deviation across three random seeds. Wave and Navier–Stokes use

64\times 64

grids; Spherical Shallow Water uses

64\times 128

. A fixed random permutation of grid locations is generated for each dataset, and nested subsets are taken without replacement using the same procedure. Masks are shared across training seeds and test trajectories. The visible count is

\lfloor v_{\mathrm{obs}}HW\rfloor

for visible fraction

v_{\mathrm{obs}}

. Standard deviations reflect training-seed variation.

Visible ratio	Wave (SMORE-Koopman, $p=100$ )	Navier–Stokes (SMORE-LRLQ, $p=72$ )	Shallow Water (SMORE-Koopman, $p=100$ )
$100\%$	$0.170\pm 0.005$	$8.64\pm 0.48$	$8.20\pm 0.35$
$50\%$	$0.114\pm 0.005$	$6.04\pm 0.21$	$5.58\pm 0.22$
$10\%$	$0.0400\pm 0.0026$	$2.34\pm 0.16$	$1.92\pm 0.10$
$4\%$	$0.0104\pm 0.0006$	$0.965\pm 0.088$	$0.696\pm 0.025$
$2\%$	$0$	$0.102\pm 0.004$	$0.177\pm 0.016$
$1\%$	$0$	$0$	$0$

Appendix

Table 13: Observability constant

c_{\mathcal{S}}=\sigma_{\min}(J_{\mathcal{S}})

for the sparse-initialization masks of Table 12 . Since the decoder is affine in the latent code,

c_{\mathcal{S}}

equals the smallest singular value of the

s\times p

matrix whose rows are the single-component decoder Jacobians

J_{0}(\mathbf{x})

at the

s

observed locations ( Lemma 3 ). Values are mean

\pm

sample standard deviation across the same three training seeds and use the same masks as Table 12 . Entries equal to zero are exact and occur when

s<p

, in which case the initial latent state is not identifiable from the observations. Values are not comparable across datasets, since the scale of

J_{0}

depends on the trained decoder and the field normalization.

PDE Surrogate Modeling

PDE surrogate modeling builds fast approximations of partial differential equation solutions for simulation, prediction, and design.

See all 322 papers

Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators
May 11, 2026Xiaoxiao Lu, Ye Yuan, Jiahao ShiLatent Dynamics Modeling PDE Surrogate Modeling
School of AIA, Huazhong University of Science and Technology
SGNO: Spectral Generator Neural Operators for Stable Long Horizon PDE Rollouts
Feb 21, 2026Jiayi Li, Penghao Jiang, Hira Saleem +3PDE Surrogate Modeling Neural Operators
A comparative study of accuracy and rollout stability of temporal surrogate models
May 24, 2026Rajarshi BiswasNeural Surrogate Modeling Long-Term Time Series Forecasting
Cargill Inc. Wayzata, MN 55391, USA

May 11, 2026J/K move · Enter open · S save

Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators

Xiaoxiao Lu, Ye Yuan, Jiahao Shi

School of AIA, Huazhong University of Science and Technology

Latent Dynamics Modeling PDE Surrogate Modeling PDE Operator Learning Spatiotemporal Forecasting Spectral Methods

Open paper AskPDF arXiv

Long-horizon forecasting of time-dependent partial differential equations (PDEs) is critical for characterizing the sustained evolution of physical systems. While neural operators have emerged as efficient surrogates, they typically learn implicit finite-time transitions from discrete observations. When deployed autoregressively, such propagators often suffer from rapid error accumulation and dynamic drift. To address this, we propose a neural forecasting framework that reformulates PDE rollout as learning a Structured Spectral Propagator (SSP) in a propagation-oriented latent space. Following an analysis-propagation-synthesis design, our framework: (i) maps physical states into a shared, time-consistent spatial representation; (ii) projects this space into a compact propagation state to isolate recurrent dynamics from fine-grained spatial details, thereby decoupling reconstruction fidelity from rollout regularity; and (iii) evolves retained spectral modes using a frequency-conditioned linear backbone complemented by a nonlinear spectral closure to account for truncated interactions. This explicit structuring endows the propagator with a strong inductive bias for coherent modal evolution. Extensive experiments demonstrate that SSP significantly outperforms state-of-the-art baselines, reducing relative

L_2

errors by up to 48.9% and exhibiting improved stability in temporal extrapolation beyond the supervised horizon.

Abstract

PDE Surrogate Modeling

Explore similar work

Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators

SGNO: Spectral Generator Neural Operators for Stable Long Horizon PDE Rollouts

A comparative study of accuracy and rollout stability of temporal surrogate models