OCL-PDE: A Generative Framework for PDE Inverse Problems with Observation-Complementary Latents

Ding Yang; Chuqi Chen; Chang Ma; Yang Xiang

cs.LGOct 5, 2026

OCL-PDE: A Generative Framework for PDE Inverse Problems with Observation-Complementary Latents

Authors: Ding Yang, Chuqi Chen, Chang Ma, Yang Xiang

Organizations: Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong · Department of Mathematics, University of Michigan, Ann Arbor, USA · Department of Materials, The Hong Kong University of Science and Technology, Hong Kong · Algorithms of Machine Learning and Autonomous Driving Research Lab, HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute, Shenzhen, China

Abstract

Partial differential equation (PDE) inverse problems are often ill-posed, making fine-scale details difficult to recover. We address this problem by introducing a learned observation-complementary latent representation that preserves reconstruction-relevant information and is combined with the observation to reconstruct the unknown field. Building on this representation, we propose OCL-PDE, a generative framework that encourages the observation to guide large-scale structure and the latent to supply complementary fine-scale details. OCL-PDE is built on a physics-aware autoencoder (AE) and conditional Flow Matching, supporting inverse reconstruction as well as forward PDE prediction. Experiments demonstrate improved reconstruction accuracy and fine-detail recovery compared with the evaluated baselines.

Figures & tables

Figure 1: Overview of OCL-PDE. The framework learns observation-complementary latents and generates them conditioned on observations to recover unknown fields. (A) The physics-aware autoencoder predicts the PDE response $\widehat{u}_{c}$ and encodes observation-complementary information in a latent $z$ ; the decoder combines $\widehat{u}_{c}$ and latent $z$ to reconstruct the input field. (B) With the autoencoder frozen, the conditional Flow learns to transform Gaussian noise into the encoder’s latents $z$ guided by the observed response. (C) At inference, the Flow generates a latent from noise conditioned on the available observation $y$ , and the decoder combines the generated latent $\widehat{z}$ and $y$ to recover the unknown field $c$ .

Figure 2: Allen–Cahn evolution of a trajectory from $t=0$ to $t=4$ .

Method	$\phi_{0}$ relative $L_{2}$ mean $\pm$ std.	Terminal-field relative $L_{2}$ at $t=4$ ; mean $\pm$ std.	Test time (s/field)
PINO ( Li et al., 2024 )	$0.18496\pm 0.04341$	$0.03110\pm 0.01316$	–
DGenNO ( Zang and Koutsourelakis, 2025 ) ( $d=512$ )	$0.66444\pm 0.33967$	$0.62601\pm 0.20541$	$73.51$
Conditional EDM+PDE ( Karras et al., 2022 )	$0.41914\pm 0.05692$	$0.41826\pm 0.07390$	$12.42$
DPS (L2 guidance) ( Chung et al., 2023 )	$0.45889\pm 0.14540$	$0.18284\pm 0.10136$	$27.56$
No- $Z$ ablation	$0.09827\pm 0.01863$	$0.01679\pm 0.01108$	–
OCL-PDE	$\mathbf{0.05897\pm 0.01817}$	$\mathbf{0.01386\pm 0.00547}$	$0.9150$

Table 1: Allen–Cahn initial-field and terminal-field relative

L_{2}

errors on 20 held-out trajectories, with test time per field.

Figure 3: Allen–Cahn initial-field recovery. Rows show reconstructions, absolute initial-field errors, spectral-method evolutions to $t=4$ , and absolute terminal-field errors. We report initial-field and terminal-field errors in relative $L_{2}$ .

Method	Noise $\rho=1\%$				Noise $\rho=3\%$
	$a$ r0	$a$ r20	$u$ r0	$u$ r20	$a$ r0	$a$ r20	$u$ r0	$u$ r20
PINO	$0.03836$	$0.03669$	$0.00452$	$0.00450$	$0.05252$	$0.03694$	$0.00580$	$0.00456$
Conditional EDM+PDE	$0.09808$	$0.09912$	$0.02349$	$0.02398$	$0.09927$	$0.09893$	$0.02372$	$0.02400$
No- $Z$ ablation	$0.03605$	$0.03151$	$0.00345$	$0.00318$	$\mathbf{0.05127}$	$0.04034$	$0.00531$	$0.00409$
OCL-PDE	$\mathbf{0.03456}$	$\mathbf{0.02249}$	$\mathbf{0.00310}$	$\mathbf{0.00265}$	$0.05239$	$\mathbf{0.02381}$	$\mathbf{0.00520}$	$\mathbf{0.00285}$

Table 2: GRF coefficient reconstruction on 16 test fields with

50\%

clean-anchor coverage: mean relative

L_{2}

errors of the recovered coefficient (

a

) and corresponding pressure (

u

), before (r0) and after (r20) pressure correction. Pressure is computed from each recovered coefficient using conservative finite differences.

Figure 4: Continuous coefficient recovery before and after pressure correction at $3\%$ noise and $50\%$ clean-anchor coverage. We show the pressure observation, reference coefficient, reconstructions, and absolute coefficient errors, and report coefficient reconstruction errors in relative $L_{2}$ .

Method	2% (318 points)			5% (794 points)			10% (1588 points)
	Hard error $\downarrow$	$u$ rel. $L_{2}$ $\downarrow$	Time	Hard error $\downarrow$	$u$ rel. $L_{2}$ $\downarrow$	Time	Hard error $\downarrow$	$u$ rel. $L_{2}$ $\downarrow$	Time
PINO	$0.06800$	$0.08264$	-	$0.05070$	$0.06317$	-	$0.04572$	$0.05456$	-
DGenNO ( $d=1024$ )	$0.13957$	$0.10923$	$99.02$	$0.14084$	$0.11891$	$96.24$	$0.14459$	$0.12381$	$84.42$
DiffusionPDE ( Huang et al., 2024 )	$0.15472$	$0.06302$	$164.8$	$0.12350$	$0.05192$	$164.5$	$0.10744$	$0.04561$	$164.6$
No- $Z$	$0.04659$	$0.02709$	-	$0.03454$	$0.01950$	-	$0.03143$	$0.01753$	-
OCL-PDE	$\mathbf{0.03524}$	$\mathbf{0.01919}$	$0.3417$	$\mathbf{0.02357}$	$\mathbf{0.01199}$	$0.3437$	$\mathbf{0.02105}$	$\mathbf{0.01086}$	$0.3440$

Table 3: Binary coeficient reconstruction on 64 test fields: hard pixel error of the recovered coefficient, relative

L_{2}

error of the corresponding pressure, and inference time (seconds per field). Pressure is computed from each recovered hard coefficient using conservative finite differences.

Figure 5: Binary coefficient recovery on the same held-out field at $2\%$ , $5\%$ , and $10\%$ observation rates (top to bottom). Percentages below reconstructions are hard pixel errors.

Figure 6: Allen–Cahn reconstructions for one pair of samples, swapping $y$ or $z$ between rows or setting either input to zero. Errors are relative $L_{2}$ against each row’s ground truth.

Figure 7: GRF Darcy latent correction in $\log a$ (Sample 0). The last three columns show what the base ( $z=0$ ) misses, what the latent adds, and what remains unrecovered, respectively. Coefficient errors are relative $L_{2}$ .

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Task	Spatial condition supplied to $C$	Latent/velocity shape	Scalar inputs
Allen–Cahn	$\eta_{\mathrm{AC}}=[\phi_{4},\mathbf{1}]$ on $128^{2}$	$8\times 8\times 8$	$\tau$ and $t_{\mathrm{cond}}=4$
GRF Darcy, no anchors	$\eta_{\mathrm{low}}=[u/s_{u},M]$ on $32^{2}$	$4\times 16\times 16$	$\tau$ only
GRF Darcy, clean anchors	$\eta_{\mathrm{MF}}=[u/s_{u},M,M_{H}]$ on $32^{2}$	$4\times 16\times 16$	$\tau$ only
Binary Darcy	$\eta_{\mathrm{bin}}=[y_{q}/s_{u},M_{q}]$ on $128^{2}$	$4\times 16\times 16$	$\tau$ only

Appendix

Table 4: Spatial inputs to the Flow condition extractor

C

. Here

\phi

is the Allen–Cahn solution and

\phi_{4}=\phi(\boldsymbol{x},4)

is its observed terminal field (

y

);

\mathbf{1}

is the all-one validity mask. For GRF Darcy,

u

denotes the native

32^{2}

pressure observation; for Binary Darcy,

y_{q}

is the raw zero-filled

128^{2}

pressure tensor. The quantity

s_{u}

is the training-set pressure standard deviation. In the clean-anchor row, selected noisy values in

u

have been replaced with clean anchors. The masks

M

and

M_{q}

mark observed pressure nodes, and

M_{H}

marks clean anchors. In GRF Darcy, Flow receives native-grid observations. Flow time

\tau

and physical time

t_{\mathrm{cond}}

are separate scalar inputs; the noise level

\rho

is not supplied to any Flow.

Task	Field grid	Latent $C\times H\times W$	AE	Log-var	Flow	Total
Allen–Cahn	$128\times 128$	$8\times 8\times 8$	30,650,596	72	22,080,456	52,731,124
GRF Darcy	$128\times 128$	$4\times 16\times 16$	30,313,272	20	10,417,444	40,730,716
Binary Darcy	$128\times 128$	$4\times 16\times 16$	30,313,272	20	10,503,668	40,816,960

Appendix

Table 5: Audited parameter counts for the models used in formal inference. The GRF Flow with clean anchors has one additional clean-anchor-mask condition channel.

Method	$d$	Initial $\phi_{0}$	Learned response	Solver response	Time (s)
DGenNO	512	$1.57088\pm 0.51192$	$0.36875\pm 0.20238$	$0.67583\pm 0.33643$	$16.67510\pm 3.37500$

Appendix

Table 6: Paper-aligned Allen–Cahn task adaptation (

N=20

). Errors are relative

L_{2}

. The discrepancy between learned and independently solved terminal responses indicates a forward-surrogate limitation.

Method	$d$	$\rho$	Coefficient $a$	Solver- $u$	Time (s)
DGenNO	1024	0	$0.71855\pm 0.26673$	$0.19793\pm 0.08747$	$24.01364\pm 3.89045$
DGenNO	1024	0.01	$0.66729\pm 0.25667$	$0.20109\pm 0.10271$	$24.22199\pm 5.68484$
DGenNO	1024	0.03	$0.71316\pm 0.24072$	$0.20244\pm 0.08668$	$25.00300\pm 4.82461$

Appendix

Table 7: Continuous Darcy results using the original DGenNO optimization procedure on the coarse pressure grid (

N=64

per setting). Errors are relative

L_{2}

Method	0% (0)	5% (45)	10% (90)	15% (135)	20% (180)
PINO	$0.04029\rightarrow 0.04558$	$0.04012\rightarrow 0.04403$	$0.04009\rightarrow 0.04267$	$0.03989\rightarrow 0.04064$	$0.03960\rightarrow 0.03944$
Conditional EDM+PDE	$0.10212\rightarrow 0.11001$	$0.10014\rightarrow 0.10678$	$0.09885\rightarrow 0.10398$	$0.09716\rightarrow 0.10090$	$0.09528\rightarrow 0.09776$
No- $Z$ ablation	$0.04013\rightarrow 0.04370$	$0.03992\rightarrow\mathbf{0.04182}$	$0.03968\rightarrow 0.04034$	$0.03902\rightarrow 0.03698$	$0.03863\rightarrow 0.03530$
OCL-PDE	$0.03967\rightarrow\mathbf{0.04297}$	$0.03934\rightarrow 0.04217$	$0.03910\rightarrow\mathbf{0.03884}$	$0.03850\rightarrow\mathbf{0.03412}$	$0.03798\rightarrow\mathbf{0.03027}$

Appendix

Table 8: GRF coefficient relative

L_{2}

\rho=1\%

with 0–20% clean-anchor coverage. Each cell reports round 0

\rightarrow

round 20; parentheses give the number of clean anchors. Lower is better.

Method	25% (225)	30% (270)	35% (315)	40% (360)	45% (405)	50% (450)
PINO	$0.03946\rightarrow 0.03874$	$0.03936\rightarrow 0.03784$	$0.03914\rightarrow 0.03729$	$0.03892\rightarrow 0.03705$	$0.03875\rightarrow 0.03686$	$0.03836\rightarrow 0.03669$
Conditional EDM+PDE	$0.09441\rightarrow 0.09654$	$0.09468\rightarrow 0.09653$	$0.09486\rightarrow 0.09633$	$0.09525\rightarrow 0.09651$	$0.09647\rightarrow 0.09754$	$0.09808\rightarrow 0.09912$
No- $Z$ ablation	$0.03823\rightarrow 0.03436$	$0.03790\rightarrow 0.03303$	$0.03738\rightarrow 0.03221$	$0.03692\rightarrow 0.03192$	$0.03670\rightarrow 0.03168$	$0.03605\rightarrow 0.03151$
OCL-PDE	$0.03756\rightarrow\mathbf{0.02833}$	$0.03727\rightarrow\mathbf{0.02549}$	$0.03663\rightarrow\mathbf{0.02360}$	$0.03581\rightarrow\mathbf{0.02324}$	$0.03527\rightarrow\mathbf{0.02271}$	$0.03456\rightarrow\mathbf{0.02249}$

Appendix

Table 9: GRF coefficient relative

L_{2}

\rho=1\%

with 25–50% clean-anchor coverage. Each cell reports round 0

\rightarrow

round 20; parentheses give the number of clean anchors. Lower is better.

Method	0% (0)	5% (45)	10% (90)	15% (135)	20% (180)
PINO	$0.06365\rightarrow 0.06083$	$0.06292\rightarrow\mathbf{0.05531}$	$0.06223\rightarrow\mathbf{0.05116}$	$0.06117\rightarrow 0.04530$	$0.05990\rightarrow 0.04240$
Conditional EDM+PDE	$0.10497\rightarrow 0.12520$	$0.10276\rightarrow 0.11878$	$0.10129\rightarrow 0.11278$	$0.09977\rightarrow 0.10632$	$0.09779\rightarrow 0.10037$
No- $Z$ ablation	$0.05806\rightarrow\mathbf{0.06051}$	$0.05780\rightarrow 0.05700$	$0.05741\rightarrow 0.05361$	$0.05668\rightarrow 0.04975$	$0.05594\rightarrow 0.04705$
OCL-PDE	$0.06092\rightarrow 0.06599$	$0.06041\rightarrow 0.06072$	$0.05994\rightarrow 0.05442$	$0.05927\rightarrow\mathbf{0.04451}$	$0.05827\rightarrow\mathbf{0.03895}$

Appendix

Table 10: GRF coefficient relative

L_{2}

\rho=3\%

with 0–20% clean-anchor coverage. Each cell reports round 0

\rightarrow

round 20; parentheses give the number of clean anchors. Lower is better.

Method	25% (225)	30% (270)	35% (315)	40% (360)	45% (405)	50% (450)
PINO	$0.05858\rightarrow 0.04053$	$0.05782\rightarrow 0.03887$	$0.05667\rightarrow 0.03791$	$0.05548\rightarrow 0.03750$	$0.05395\rightarrow 0.03727$	$0.05252\rightarrow 0.03694$
Conditional EDM+PDE	$0.09668\rightarrow 0.09789$	$0.09680\rightarrow 0.09735$	$0.09676\rightarrow 0.09642$	$0.09698\rightarrow 0.09636$	$0.09812\rightarrow 0.09739$	$0.09927\rightarrow 0.09893$
No- $Z$ ablation	$0.05516\rightarrow 0.04560$	$0.05487\rightarrow 0.04344$	$0.05411\rightarrow 0.04221$	$0.05316\rightarrow 0.04156$	$0.05204\rightarrow 0.04085$	$0.05127\rightarrow 0.04034$
OCL-PDE	$0.05731\rightarrow\mathbf{0.03597}$	$0.05691\rightarrow\mathbf{0.03019}$	$0.05625\rightarrow\mathbf{0.02633}$	$0.05488\rightarrow\mathbf{0.02566}$	$0.05317\rightarrow\mathbf{0.02457}$	$0.05239\rightarrow\mathbf{0.02381}$

Appendix

Table 11: GRF coefficient relative

L_{2}

\rho=3\%

with 25–50% clean-anchor coverage. Each cell reports round 0

\rightarrow

round 20; parentheses give the number of clean anchors. Lower is better.

Method	$a$ relative $L_{2}$			Solver- $u$ relative $L_{2}$			Mean time (s/field)
	$\rho=0$	$\rho=0.01$	$\rho=0.03$	$\rho=0$	$\rho=0.01$	$\rho=0.03$
PINO	$0.03561$	$0.03986$	$0.06239$	$0.00430$	$0.00471$	$0.00711$	$0.00141$
DGenNO ( $d=1024$ )	$0.15056$	$0.14958$	$0.15234$	$0.03358$	$0.03316$	$0.03405$	$99.74$
Conditional EDM+PDE	$0.09575$	$0.09636$	$0.09943$	$0.02633$	$0.02637$	$0.02674$	$12.25$
No- $Z$ ablation	$0.02742$	$0.03858$	$\mathbf{0.05704}$	$0.00318$	$0.00371$	$0.00629$	$0.01571$
OCL-PDE	$\mathbf{0.01807}$	$\mathbf{0.03686}$	$0.05740$	$\mathbf{0.00192}$	$\mathbf{0.00334}$	$\mathbf{0.00620}$	$2.541$

Appendix

Table 12: Robustness of continuous Darcy recovery to additive pressure noise on the aligned

32\times 32

pressure grid. Independent Gaussian noise with standard deviation

\rho s_{u}

is added to all interior pressure values; the boundary remains zero. The case

\rho=0

is noise-free. Reported coefficient and solver-

u

errors are means over 64 held-out fields. For each method with recorded timing, test time is the mean per-field inference time over all three noise levels, excluding numerical Darcy solves; “–” denotes unavailable timing. Solver-

u

denotes an independent numerical Darcy solve from each reconstructed coefficient. Lower is better.

Method	2% (318 points)			5% (794 points)			10% (1588 points)
	Boundary F1 $\uparrow$	ASSD (px) $\downarrow$	HD95 (px) $\downarrow$	Boundary F1 $\uparrow$	ASSD (px) $\downarrow$	HD95 (px) $\downarrow$	Boundary F1 $\uparrow$	ASSD (px) $\downarrow$	HD95 (px) $\downarrow$
PINO	$0.47774$	$3.43339$	$13.20167$	$0.60188$	$2.40210$	$9.76224$	$0.65318$	$2.12762$	$9.99703$
DGenNO ( $d=1024$ )	$0.49796$	$5.12891$	$21.74188$	$0.50092$	$5.21081$	$22.72245$	$0.49330$	$5.39692$	$22.19505$
DiffusionPDE	$0.3540$	$7.6068$	$29.0139$	$0.4270$	$6.5002$	$26.5313$	$0.4493$	$6.1458$	$25.1457$
No- $Z$	$0.6279$	$2.8237$	$13.3906$	$0.7514$	$2.0703$	$10.8046$	$0.7894$	$1.8443$	$9.4692$
OCL-PDE	$\mathbf{0.7509}$	$\mathbf{1.9840}$	$\mathbf{10.0763}$	$\mathbf{0.8721}$	$\mathbf{1.2133}$	$\mathbf{6.4314}$	$\mathbf{0.8943}$	$\mathbf{0.9803}$	$\mathbf{5.2465}$

Appendix

Table 13: Additional Binary Darcy metrics at

2\%

5\%

, and

10\%

observation rates. Boundary F1 uses a one-pixel tolerance; ASSD and HD95 are measured in pixels. Bold values are best within each observation rate.

Method	$d$	Observed	Hard error	Solver- $u$	Time (s)
DGenNO	1024	$2\%$	$0.16463\pm 0.07715$	$0.13248\pm 0.06821$	$25.21020\pm 5.67132$
DGenNO	1024	$5\%$	$0.18134\pm 0.10124$	$0.14612\pm 0.08035$	$26.16990\pm 2.63666$
DGenNO	1024	$10\%$	$0.17399\pm 0.07537$	$0.15497\pm 0.09822$	$24.58750\pm 5.65273$

Appendix

Table 14: DGenNO with the original inference procedure for Binary Darcy from random pressure observations (

N=64

per setting). Entries are mean

\pm

sample standard deviation. Both errors are reported as ratios.

Observation density	Method	Clean	80 dB	50 dB
$2\%$	PINO	$0.068003$	$0.067999$	$0.068479$
	No- $Z$	$0.046590$	$0.046590$	$0.047270$
	OCL-PDE	$\mathbf{0.035237}$	$\mathbf{0.035226}$	$\mathbf{0.036256}$
$5\%$	PINO	$0.050697$	$0.050689$	$0.050801$
	No- $Z$	$0.034540$	$0.034532$	$0.035040$
	OCL-PDE	$\mathbf{0.023565}$	$\mathbf{0.023566}$	$\mathbf{0.024655}$

Appendix

Table 15: Binary Darcy reconstruction under observation noise without retraining or refinement. Mean hard pixel errors over 64 fields are reported as fractions.

Figure 8: DiffusionPDE coefficient-observation sanity check with the main experiment’s epoch 350 EMA checkpoint. Rows show the same held-out field at $2\%$ , $5\%$ , and $10\%$ random pressure observation rates. Columns show observed pressure, the 500 fixed coefficient observations, ground truth, pressure-only recovery, and recovery with both types of observations. Reconstructed coefficients are thresholded at the phase midpoint; annotations report hard pixel error and solver- $u$ relative $L_{2}$ error from an independent numerical solve.

Figure 9: Full Allen–Cahn decoder-branch intervention for two fixed test pairs: trajectories 6 and 15 in the first two rows, and trajectories 0 and 3 in the last two rows. Columns show the reference field, matched reconstruction, exchanged observation, exchanged latent, zero latent, and zero observation. The plot labels $U$ and $Z$ denote observation $y$ and latent $z$ , respectively. Each reconstruction is annotated with its relative $L_{2}$ error.

Allen–Cahn: $n=20$ , $f=\phi(\cdot,0)$
Reconstruction	Hard error	$E_{\mathrm{physical}}$	$E_{f}$	$E_{\mathrm{hi}}$	$Q$
$D_{\theta_{D}}(z=0,y)$	–	$100.67\pm 0.62$	$100.67\pm 0.62$	$188.76\pm 23.43$	$0$
$D_{\theta_{D}}(z^{\star},y)$	–	$2.66\pm 0.72$	$2.66\pm 0.72$	$55.62\pm 3.47$	$0.9993$
$D_{\theta_{D}}(z_{flow},y)$	–	$5.90\pm 1.82$	$5.90\pm 1.82$	$78.95\pm 3.99$	$0.9964$
GRF Darcy: $n=64$ , $f=\log a$
$D_{\theta_{D}}(z=0,y)$	–	$3.66\pm 0.93$	$9.08\pm 2.51$	$97.98\pm 2.42$	$0$

Appendix

Table 16: Fixed-observation latent diagnostics. Errors are percentages (mean

\pm

sample standard deviation);

Q

is the pooled analysis-field squared-error reduction relative to

z=0

. The oracle uses the ground-truth encoder mean

\mu

. Flow uses

K=64

samples for Allen–Cahn and GRF Darcy, and

K=16

for Binary Darcy. Binary outputs are thresholded after averaging phase probabilities; hard pixel error applies only to Binary Darcy.

Figure 10: GRF Darcy latent corrections. Flow-generated reconstruction and zero-latent base, with the target residual, latent correction, and remaining error in $f=\log a$ . We report coefficient reconstruction errors in relative $L_{2}$ .

Figure 11: Allen–Cahn latent corrections. Flow-generated reconstruction and zero-latent base, with the target residual, latent correction, and remaining error in $f=\phi(\cdot,0)$ . We report initial-field reconstruction errors in relative $L_{2}$ .

Figure 12: Binary Darcy latent corrections at $5\%$ observations. Flow-generated reconstruction and zero-latent base, with the target residual, latent correction, and remaining error in $f=\log a$ . We report hard pixel errors.

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Bayesian Inverse Problems Partial Differential Equations Conditional Flow Matching

Open paper AskPDF arXiv

Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.

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