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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