Beyond Local Accuracy: Benchmarking the Rollout Reliability of PDE Foundation Models

Jaewan Park; Seid Koric; Giri Krishnan; Sudhakar Pamidighantam; Surresh Marru

cs.LGOct 22, 2026

Beyond Local Accuracy: Benchmarking the Rollout Reliability of PDE Foundation Models

Authors: Jaewan Park, Seid Koric, Giri Krishnan, Sudhakar Pamidighantam, Surresh Marru

Organizations: National Center for Supercomputing Applications (NCSA), University of Illinois Urbana–Champaign · Georgia Institute of Technology

Abstract

PDE foundation models enable data-efficient adaptation but are deployed autoregressively, so short-horizon accuracy need not imply reliable simulation. We ask whether the short-horizon advantage of a foundation-model recipe persists when its predictions are repeatedly fed back as inputs for a long horizon. We define error amplification as G=E(H_f)/E(H_s), which measures full-horizon error relative to short-horizon error. We evaluate seven models on seven two-dimensional time-dependent PDE datasets: scratch-trained FNO and DeepONet; fine-tuned DPOT, GPhyT, Poseidon, and Walrus; and frozen in-context VICON. We report VRMSE, relative L_2, and relative MAE. Six datasets provide paired short- and full-horizon scores, while CE-RM contributes short-rollout sweeps only. At least one fine-tuned foundation model outperforms both scratch baselines in 26 of 30 short-rollout dataset--budget cells under every metric. Nevertheless, full-horizon VRMSE exceeds short-horizon VRMSE in 140 of 144 paired trained configurations, with a median error amplification of $2.04$ . The count below VRMSE one falls from 104 to 53, and 12 full rollouts exceed ten although none does at the short horizon. Walrus achieves the lowest full-horizon VRMSE in 16 of 24 cells, yet its median error amplification is $2.70$ . Relative L_2 and relative MAE show the same qualitative pattern. These results show that favorable short-horizon rankings can coexist with substantial error amplification. PDE foundation-model benchmarks should therefore report horizon-resolved absolute errors, amplification statistics, and failure incidence rather than a single local forecast score.

Figures & tables

Dataset	Family	Grid	Physical channels	Frames	Test traj.	$N$ sweep	Horizon coverage
FSI	fluid–structure interaction	$128^{2}$	$(u,v,p)$	101	10	${1,4,16,32}$	short, full
Cylinder	cylinder flow	$128\times 256$	$(u,v,p)$	101	10	${1,4,16,64}$	short, full
Controlled cylinder	actuated cylinder flow	$128\times 256$	$(u,v,p)$	101	10	${1,4,16,64}$	short, full
Foil	flow past an airfoil	$128\times 256$	$(u,v,p)$	101	10	${1,4,16,64}$	short, full
DR	reaction–diffusion	$128^{2}$	activator, inhibitor	101	100	${1,4,16,64}$	short, full
CondNS	conditioned incompressible NS	$128^{2}$	scalar, $v_{x}$ , $v_{y}$	56	130	${1,4,16,64}$	short, full

Table 1: The seven benchmark datasets. All datasets provide VRMSE, relative

L_{2}

, and relative MAE. The first six provide short- and full-horizon scores, whereas CE-RM provides the short-horizon track only. RealPDEBench entries use simulated fields; no experimental measurements are evaluated.

Model	Class	Parameters	Published pretraining source	Adaptation here	Track
FNO	neural operator	101M	none	from scratch at each $N$	swept
DeepONet	neural operator	68–70M	none	from scratch at each $N$	swept
DPOT-L	pretrained operator	509M	PDEBench, PDEArena, CFDBench	fine-tuned at each $N$	swept
GPhyT-L	physics transformer	385M	The Well and in-house simulations	fine-tuned at each $N$	swept
Poseidon-L	pretrained operator transformer	629M	PDEGym	fine-tuned at each $N$	swept
Walrus	continuum foundation model	1.3B	19 continuum scenarios	fine-tuned at each $N$	swept

Table 2: Models and adaptation regimes. Parameter counts correspond to the configurations evaluated in this benchmark and are approximate. The six swept models produce one trained checkpoint at every available trajectory budget; VICON is a frozen in-context reference.

Panel of Figure 1 — Figure 1: Short- versus full-horizon errors on the six datasets with paired scoring intervals: (a) VRMSE, (b) relative $L_{2}$ , and (c) RMAE. Each trained-model point pairs the same dataset, budget $N$ , and checkpoint; the six VICON points are frozen in-context references. The dashed diagonal marks equal error, and the dotted guides mark error one. CE-RM is excluded because it has no full-horizon score. The full-horizon axis ends at $10^{1}$ ; respectively 12, 11, and 12 values in panels (a)–(c) are off-scale (diverged) and reported in the appendices.

Metric	FM envelope better (S/F), out of 30/24	Winner retained	Paired $E<1$ (S/F), out of 144/144	Paired $E>10$ (S/F), out of 144/144	Pairwise reversals
VRMSE	$26/21$	$18/24$	$104/53$	$0/12$	$104/360$
Relative $L_{2}$	$26/21$	$16/24$	$121/81$	$0/11$	$104/360$
Relative MAE	$26/21$	$19/24$	$121/80$	$0/12$	$108/360$

Table 3: Aggregate diagnostics for the six swept models. S/F denotes short/full scoring intervals. The short FM-envelope count includes all seven datasets; its full count uses the six datasets with paired horizons. All other horizon comparisons use 24 paired dataset–budget cells and 144 trained-model configurations. Pairwise reversals are measured over

24\binom{6}{2}=360

comparisons.

Panel of Figure 2 — Figure 2: Short-horizon VRMSE versus the number of training trajectories for the six trained models on all seven datasets. Each model retains its native temporal configuration. FSI through CondNS are scored over frames 10–20; CE-RM uses the model-specific short rollout defined in Section 3 . Frozen VICON is omitted because it is $N$ -independent.

Dataset	Report	$N$	FNO	DeepONet	DPOT	GPhyT	Poseidon	Walrus
FSI	VRMSE S/F	32	0.3800/1.1699	0.6279/1.2005	0.1963/0.7915	0.1904/1.5338	0.3276/0.8857	0.1093/0.7401
Cylinder	VRMSE S/F	64	0.0043/0.0147	0.0425/0.0819	0.0393/0.4999	0.0054/0.6907	0.1984/0.7187	0.0359/0.5678
Controlled cylinder	VRMSE S/F	64	0.2768/1.0866	0.4875/1.1184	0.0679/0.6161	0.0650/1.0230	0.3405/1.2643	0.0771/0.8203
Foil	VRMSE S/F	64	0.3975/0.8823	0.5605/0.9875	0.2513/0.6870	0.2349/0.8261	0.6138/0.7450	0.2042/0.7104
DR	VRMSE S/F	64	0.2104/0.4113	1.2155/1.2340	0.0233/0.0298	0.0301/0.0631	0.1161/0.5338	0.0258/0.0612
CondNS	VRMSE S/F	64	0.4546/0.7897	0.7396/1.2216	0.2545/0.6001	0.4455/0.9285	0.6850/0.9155	0.1483/0.5498

Table 4: Results at the largest matched trajectory (largest

N

) budget for each dataset. Entries are short/full VRMSE for the six datasets with paired horizons and short VRMSE for CE-RM. FSI uses

N=32

, the next five datasets use

N=64

, and CE-RM uses

N=1024

Dataset	$N$	FNO	DeepONet	DPOT	GPhyT	Poseidon	Walrus	VICON †
FSI	1	0.8044	1.1615	0.9176	0.4946	0.7599	0.3128	0.7677
	4	0.6284	0.7997	0.8101	0.3841	0.8025	0.2481
	16	0.5246	0.7576	0.3329	0.2693	0.4513	0.1650
	32	0.3800	0.6279	0.1963	0.1904	0.3276	0.1093
Cylinder	1	2.7614	3.0381	1.6192	1.0302	1.8800	1.1067	0.7926
	4	2.0683	3.0814	0.8167	0.5360	2.4884	0.2040

Table 5: Short-horizon VRMSE (

\downarrow

) for all seven datasets. FSI through CondNS are scored over frames 10–20; CE-RM uses each model’s native short rollout. Frozen VICON contributes one

N

-independent reference per dataset and is evaluated on its fixed

128\times 128

grid.

Dataset	Short VRMSE	Full VRMSE
FSI	0.7677	1.0767
Cylinder	0.7926	1.0818
Controlled cylinder	0.7660	1.1361
Foil	0.7773	1.0114
DR	0.9933	1.0747
CondNS	0.7832	1.1758

Table 6: Frozen in-context VICON VRMSE. The first six datasets provide paired short- and full-horizon scores; CE-RM provides a short-horizon score only. VICON is evaluated on its fixed

128\times 128

grid and does not define an

N

-sweep.

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Metric	$N$	FNO	DeepONet	DPOT	GPhyT	Poseidon	Walrus
FSI	Full VRMSE	1	1.1338	1.1857	13.7525	$5.27\times 10^{3}$	1.1485	1.0644
		4	1.1458	1.1764	18.7319	1.9471	1.2374	0.9835
		16	1.1972	1.1950	16.6726	1.7582	0.9745	0.9301
		32	1.1699	1.2005	0.7915	1.5338	0.8857	0.7401
Cylinder	Full VRMSE	1	3.1368	3.1663	2.5079	$1.05\times 10^{5}$	2.8369	1.2967
		4	2.1199	3.1016	1.7748	3.7422	3.0073	0.9177

Appendix

Table 7: Full-horizon VRMSE (

\downarrow

) for the six datasets with a full interval. RealPDEBench and DR use frames 10–100; CondNS uses frames 10–55.

Dataset	$N$	FNO	DeepONet	DPOT	GPhyT	Poseidon	Walrus
FSI	1	0.7051/0.9567	0.9850/0.9984	0.7523/11.9418	0.4321/ $(4.5\times 10^{3})$	0.6361/0.9414	0.2661/0.9251
	4	0.5014/0.9468	0.6897/0.9515	0.6693/11.2274	0.3320/1.6512	0.6787/1.0306	0.2171/0.8512
	16	0.4291/0.9907	0.6172/0.9968	0.2840/10.5008	0.2322/1.4436	0.3749/0.8001	0.1466/0.7855
	32	0.3198/0.9724	0.5050/1.0007	0.1675/0.6819	0.1655/1.2517	0.2781/0.7417	0.0956/0.6341
Cylinder	1	1.5538/1.8431	1.6995/1.8670	0.8200/1.3470	0.7195/ $(6.76\times 10^{4})$	1.1888/1.6645	0.8844/1.0123
	4	1.1264/1.1462	1.8541/1.8243	0.4598/0.9685	0.3897/2.5539	1.5138/1.7779	0.1586/0.7308

Appendix

Table 8: Relative

L_{2}

(

\downarrow

) for all seven datasets. FSI through CondNS are short/full; CE-RM entries are short only.

Dataset	$N$	FNO	DeepONet	DPOT	GPhyT	Poseidon	Walrus
FSI	1	0.6905/0.9912	1.0074/1.0425	0.8017/15.8779	0.4137/ $(2.14\times 10^{3})$	0.6039/0.9370	0.2418/0.9069
	4	0.5152/0.9550	0.6799/0.9479	0.7244/12.2328	0.3176/1.7265	0.7137/1.0956	0.2040/0.8476
	16	0.4509/1.0127	0.6295/1.0088	0.2807/10.6968	0.2238/1.4445	0.3763/0.7745	0.1418/0.7395
	32	0.3452/1.0051	0.5133/0.9863	0.1621/0.6702	0.1647/1.2636	0.2836/0.7068	0.0907/0.5895
Cylinder	1	1.5089/1.7577	1.6370/1.7790	0.8181/1.4050	0.6862/ $(3.58\times 10^{4})$	1.1769/1.5955	0.7656/0.8778
	4	1.1356/1.1580	1.7660/1.7353	0.4543/0.9977	0.3824/2.5046	1.4466/1.6939	0.1178/0.6206

Appendix

Table 9: Relative MAE (

\downarrow

) for all seven datasets. FSI through CondNS are short/full; CE-RM entries are short only.

Dataset	$N$	VRMSE	rel- $L_{2}$	RMAE
FSI	–	0.7677/1.0767	0.6703/0.9002	0.6578/0.9411
Cylinder	–	0.7926/1.0818	0.6259/0.8370	0.6030/0.8157
Controlled cylinder	–	0.7660/1.1361	0.6097/0.8875	0.6042/0.9176
Foil	–	0.7773/1.0114	0.6003/0.7558	0.5270/0.7202
DR	–	0.9933/1.0747	0.9869/1.0010	0.9877/1.0009
CondNS	–	0.7832/1.1758	0.6472/0.9962	0.6344/1.0032

Appendix

Table 10: VRMSE, relative

L_{2}

, and relative MAE for frozen in-context VICON across all seven datasets. The first six entries are short/full; CE-RM entries are short only. This reference is not included in the trained-model aggregate counts of Table 3 .

Dataset	Sweep maximum	Full-pool size	VRMSE	rel- $L_{2}$	RMAE
FSI	32	36	0.0965/0.6446	0.0859/0.5462	0.0882/0.5042
Cylinder	64	not recorded	0.0214/0.3751	0.0162/0.2954	0.0120/0.2471
Controlled cylinder	64	76	0.0188/0.3711	0.0151/0.3016	0.0131/0.2602
Foil	64	79	0.1986/0.6819	0.1557/0.5633	0.0856/0.3708

Appendix

Table 11: Additional Walrus full-pool endpoints. Metrics are short/full.

Figure 3: CondNS (passive scalar) — rollout comparison. Ground truth and all seven models across frames $t=10$ – $20$ at $N=64$ . Visual sharpness tracks the reported relative- $L_{2}$ : Walrus and DPOT retain the fine filaments, whereas Poseidon, DeepONet, and VICON diffuse or coarsen.

Figure 4: CondNS (passive scalar) — data efficiency. The five swept models at $t=20$ as $N$ increases (rows), with the $N$ -independent Walrus and VICON reference columns at right.

Figure 5: FSI (pressure) — rollout comparison. Ground truth and all seven models across frames $t=10$ – $20$ at $N=32$ . The moving low-pressure cores are tracked closely by Walrus, FNO, DPOT, and Poseidon; VICON coarsens and loses structure by late frames.

Figure 6: FSI (pressure) — data efficiency. Swept models at $t=20$ versus $N$ , with Walrus and VICON reference columns. GPhyT is grainy at $N{=}1$ and sharpens with more trajectories.

Figure 7: DR (activator) — rollout comparison. Ground truth and all seven models across frames $t=10$ – $20$ at $N=64$ . DPOT, GPhyT, Poseidon, and Walrus reproduce the Turing pattern; DeepONet blurs it and VICON collapses to a near-constant field after the seeded frame.

Figure 8: DR (activator) — data efficiency. Swept models at $t=20$ versus $N$ , with Walrus and VICON reference columns. DeepONet is unstable (a flat or blobby field at several $N$ ); VICON is $N$ -independent and remains collapsed.

Figure 9: Cylinder (pressure) — rollout comparison. Ground truth and all seven models across frames $t=10$ – $20$ at $N=64$ on the native $128\times 256$ grid. The von Kármán vortex street is tracked by most trained models; VICON, which operates at $128^{2}$ and is resized to the grid for display, is diffuse.

Figure 10: Cylinder (pressure) — data efficiency. Swept models at $t=20$ versus $N$ , with Walrus and VICON reference columns.

Figure 11: Controlled cylinder (pressure) — rollout comparison. Ground truth and all seven models across frames $t=10$ – $20$ at $N=64$ ( $128\times 256$ ; VICON resized from $128^{2}$ ).

Figure 12: Controlled cylinder (pressure) — data efficiency. Swept models at $t=20$ versus $N$ , with Walrus and VICON reference columns.

Figure 13: Foil (pressure) — rollout comparison. Ground truth and all seven models across frames $t=10$ – $20$ at $N=64$ ( $128\times 256$ ; VICON resized from $128^{2}$ ).

Figure 14: Foil (pressure) — data efficiency. Swept models at $t=20$ versus $N$ , with Walrus and VICON reference columns.

Figure 15: CE-RM (density) — rollout comparison. Richtmyer–Meshkov interface for all seven models across frames $t=10$ – $20$ . FNO and DeepONet advance at $\Delta t=2$ from an early context window and reach only $t=14$ , so their later cells are marked n/a . CE-RM has no data-efficiency panel because per- $N$ rollout archives were not retained for every swept model.

Figure 16: CondNS — per-frame rollout error. All three metrics versus prediction frame at $N=64$ . Every model amplifies error monotonically and saturates by mid-rollout; Walrus stays lowest throughout and DeepONet and VICON saturate near unity.

Figure 17: FSI — per-frame rollout error. Metrics versus frame at $N=32$ . DPOT and GPhyT track Walrus most closely; VICON is nearly flat and high, reflecting its frozen in-context inference.

Figure 18: DR — per-frame rollout error. Metrics versus frame at $N=64$ . DPOT, GPhyT, and Poseidon remain low across the horizon, whereas DeepONet and VICON sit near unity (no better than predicting the mean field) from the first predicted frame.

Figure 19: Cylinder — per-frame rollout error. Metrics versus frame at $N=64$ . The periodic wake is the easiest target: most trained models stay well below $0.1$ , with VICON the clear outlier.

Figure 20: Controlled cylinder — per-frame rollout error. Metrics versus frame at $N=64$ . Walrus, DPOT, and GPhyT remain low; Poseidon and DeepONet amplify faster.

Figure 21: Foil — per-frame rollout error. Metrics versus frame at $N=64$ .

Figure 22: CE-RM — per-frame rollout error. Metrics versus frame for the representative fine-tune. Markers denote the frames each model actually predicts: FNO and DeepONet ( $\Delta t=2$ from an early window) terminate at $t=14$ , Poseidon reaches $t=20$ at $\Delta t=2$ , and the remaining models step at $\Delta t=1$ . Walrus again stays lowest throughout.

PDE Foundation Models

See all 11 papers

Aug 12, 2026

Distillation of Foundation Models for Time-dependent PDEs

Foundation models for time-dependent partial differential equations (PDEs) are trained on large and diverse collections of physical systems and can generalize effectively to new downstream tasks. After fine-tuning on only a few trajectories from a target domain, they can achieve strong accuracy in low-data regimes. However, these models are typically large and computationally intensive, limiting their usefulness as fast surrogates for numerical solvers. We propose Teacher Rollout Extension (TREX), a knowledge distillation framework that transfers the predictive capability of a pretrained foundation model into a compact and efficient student. Starting from a fine-tuned teacher, TREX augments limited downstream data by generating long synthetic trajectories through teacher rollouts, optionally with periodic noise injection. This procedure samples from the teacher-induced rollout distribution without requiring explicit knowledge of the initial-condition distribution, while exposing the student to long-horizon states and local recovery behavior around states encountered during autoregressive prediction. The student can further incorporate task-specific inductive biases, such as equivariance, that the teacher does not necessarily enforce. We evaluate TREX on multiple PDE benchmarks. The resulting students can match or surpass the teacher's accuracy while reducing the number of parameters by several orders of magnitude and achieving more than an order-of-magnitude speedup in inference.

Daniel Musekamp, Boshra Ariguib, Andrei Manolache +1

University of Stuttgart · University of Cologne · Bitdefender

Efficient Neural Network Inference PDE Surrogate Modeling PDE Foundation Models Knowledge Distillation

Ask PDF

May 21, 2026

ARC-STAR: Auditable Post-Hoc Correction for PDE Foundation Models

Partial differential equation (PDE) foundation models are pretrained networks that forecast how physical fields like velocity and pressure evolve from a single reusable solver. On unfamiliar flows their predictions drift step by step, errors concentrate in a few regions, yet retraining destabilizes the network and uniform post-hoc correction overlooks this spatial concentration. To address this, we propose a frozen-solver post-hoc correction framework, Adaptive Risk-Calibrated Spatial Triage for Auditable Refinement (ARC-STAR). ARC-STAR organizes correction into three stages: a global corrector removes broad solver bias, a blockwise local refiner cleans the post-global residual, and, at deployment, a label-free score routes refinement to high-risk blocks under a compute budget. The framework is designed to be (i) frozen-host, preserving the pretrained solver without fine-tuning; (ii) auditable, with global and local stages trained and evaluated separately for measurable contributions; and (iii) budget-aware, using a blockwise interface that either refines the full field or routes limited compute to high-risk regions. Across five flow benchmarks spanning ten regime cells, ARC-STAR is the only method that cuts velocity rollout error by at least 36x over raw Poseidon on every cell. The global stage reduces raw host error by 91-99%, and the local stage further reduces the remaining post-global residual by up to 94.4%.

Chengze Li, Lingwei Wei, Li Sun +7

University of Illinois Chicago · 2Beijing University of Posts and Telecommunications · 3North China Electric Power University

PDE Surrogate Modeling PDE Foundation Models Computational Fluid Dynamics

Ask PDF

Mar 2, 2026

Probabilistic Retrofitting of Learned Simulators

Dominant approaches for modelling Partial Differential Equations (PDEs) rely on deterministic predictions, yet many physical systems of interest are inherently chaotic and uncertain. While training probabilistic models from scratch is possible, it is computationally expensive and fails to leverage the significant resources already invested in high-performing deterministic backbones. In this work, we adopt a training-efficient strategy to transform pre-trained deterministic models into probabilistic ones via retrofitting with a proper scoring rule: the Continuous Ranked Probability Score (CRPS). Crucially, this approach is architecture-agnostic: it applies the same adaptation mechanism across distinct model backbones with minimal code modifications. The method proves highly effective across different scales of pre-training: for models trained on single dynamical systems, we achieve 20-54% reductions in rollout CRPS and up to 30% improvements in variance-normalised RMSE (VRMSE) relative to compute-matched deterministic fine-tuning. We further validate our approach on a PDE foundation model, trained on multiple systems and retrofitted on the dataset of interest, to show that our probabilistic adaptation yields an improvement of up to 40% in CRPS and up to 15% in VRMSE compared to deterministic fine-tuning. Validated across diverse architectures and dynamics, our results show that probabilistic PDE modelling need not require retraining from scratch, but can be unlocked from existing deterministic backbones with modest additional training cost.

Cristiana Diaconu, Miles Cranmer, Richard E. Turner +2

The Polymathic AI Collaboration · University of Cambridge · Alan Turing Institute

Surrogate Modeling PDE Surrogate Modeling

Ask PDF

Abstract

PDE Foundation Models

Explore similar work

Distillation of Foundation Models for Time-dependent PDEs

ARC-STAR: Auditable Post-Hoc Correction for PDE Foundation Models

Probabilistic Retrofitting of Learned Simulators