Sparsely-Supervised Data Assimilation via Physics-Informed Schrödinger Bridge
Authors: Dohyun Bu, Chanho Kim, Seokun Choi, Jong-Seok Lee
Organizations: Department of Industrial & Systems Engineering, KAIST, Daejeon, Repulic of Korea · Department of Mathematical Sciences, KAIST, Daejeon, Repulic of Korea
Data assimilation (DA) for systems governed by partial differential equations (PDE) aims to reconstruct full spatiotemporal fields from sparse high-fidelity (HF) observations while respecting physical constraints. While full-grid low-fidelity (LF) simulations provide informative priors in multi-fidelity settings, recovering an HF field consistent with both sparse observations and the governing PDE typically requires per-instance test-time optimization, which becomes a major bottleneck in time-critical applications. To alleviate this, amortized reconstruction using generative models has recently been proposed; however, such approaches rely on full-field HF supervision during training, which is often impractical in real-world settings. From a more realistic perspective, we propose the Physics-Informed Conditional Schrödinger Bridge (PICSB), which transports an informative LF prior toward an observation-conditioned HF posterior without any additional inference-time guidance. To enable learning without HF endpoints, PICSB employs an iterative surrogate-endpoint refresh scheme, and directly incorporates PDE residuals into the training objective while enforcing observations via hard conditioning throughout sampling. Experiments on fluid PDE benchmarks demonstrate that PICSB enables extremely fast spatiotemporal field reconstruction while maintaining competitive accuracy under sparse HF supervision.
Figures & tables
Method
Zero-shot
DA
Label-Efficient
Guidance-free
PINNs ( Raissi et al., 2019 )
✗
✓
✓
✓
MMGN ( Luo et al., 2024 )
✗
✓
✓
✗
FNO, NOICL ( Li et al., 2021 ; Chen et al., 2024 )
✓
✗
✗
✗
P 2 INN ( Cho et al., 2024 )
✓
✗
✓
✓
DiffusionPDE, ECI, PCFM
✓
✓
✗
✗
( Huang et al., 2024 ; Cheng et al., 2025 ; Utkarsh et al., 2025 )
Table 1: Comparison of learning-based field reconstruction methods for PDE-governed systems. Zero-shot indicates amortized reconstruction without test-time re-optimization. DA indicates the capability to solve an observation-conditioned inverse problem using sparse HF observations. Label-Efficient indicates training under sparse HF supervision, where HF targets are available only at partial spatiotemporal locations. Guidance-free indicates that PDE residuals are incorporated directly into training, avoiding inference-time correction.
Figure 1: Overview of PICSB training with sparse supervision. The static model (parameters θ~ ) provides surrogate endpoints to construct the bridge reference, while the trainable sampler (parameters θ ) is updated by minimizing the physics residual under hard observation projection at every sampling step. Here, with refresh period C=20 , iteration 73 uses a static model last refreshed at iteration 60 .
Figure 2: Qualitative comparison of reconstructed fields under ( R3 ). For Burgers and Darcy flow, a single test sample is shown. For Kolmogorov, a snapshot at γ=0.64 from the time horizon γ∈(0,1.25] is visualized. LF, sparse HF observations (obs.), PINNs, Guidance, and PICSB are compared against the HF reference (ref.). Zoomed-in regions highlight local reconstruction quality.
PDE
Task
PINNs
Guidance
PICSB
Burgers
R1
8.29
0.99
2.59
R2
13.48
19.04
14.05
R3
11.79
10.66
14.25
Darcy
R1
5.69
12.02
0.29
R3
5.64
27.39
0.28
Kolmogorov
R1
64.47
8.99
7.32
Table 2: Relative reconstruction error (in %) on the test set, reported as RelError in Eq. ( 15 ) under each observation regime ( R1 – R3 ). All values are averaged over the four test instances per PDE benchmark. For each task, the best-performing method is underlined , and the overall best result within each PDE benchmark is highlighted in bold .
PDE
PINNs
Guidance
PICSB
Burgers
1733
174
0.023
Darcy
2682
173
0.023
Kolmogorov
10680
801
0.134
Table 3: Average wall-time (seconds) for inference under ( R3 ).
Appendix figures & tables28 assets
Supplementary material from the paper’s appendix.
Appendix
PDE
Enc. ch.
Dec. ch.
#Enc.
#Dec.
Kernel
dcond
Bridge-time steps
ϵ
LR
Train iters.
Burgers
128,256
256,128
2
2
3×3
8
10
10−2
10−3
100k
Darcy
128,256
256,128
2
2
3×3
8
10
10−2
10−6
100k
Kolmogorov
128,256
256,128
2
2
3×3
8
10
10−2
10−3
100k
Appendix
Table 4: PICSB architectural and training hyperparameters used for each PDE benchmark. All benchmarks share the same network architecture and training schedule, except for the input/output channel dimensionality and learning rate. The table summarizes the key structural and optimization-related hyperparameters that define the PICSB model.
PDE
Hidden dim
Depth
Ncolloc
λobs
λphys
LR
Epochs
Extra settings
Burgers
512
6
8192
1.0
1.0
10−3
100k
–
Darcy
512
6
8192
1.0
1.0
10−3
100k
λbc=1,Nbc=2048 , grad clip (1e-5)
Kolmogorov
256
4
8192
1.0
1.0
10−3
100k
–
Appendix
Table 5: PINNs architectural and optimization hyperparameters used for each PDE benchmark. The table summarizes the key structural and optimization-related hyperparameters that define the PINNs baseline, with Darcy-specific stabilization settings reported in the last column.
PDE
Sampling steps
σmin
σmax
ρ
λobs
λphys
λbc
Burgers
2000
0.002
80
7
320
100
–
Darcy
2000
0.002
80
7
10−3
10−1
10−1
Kolmogorov
2000
0.002
80
7
3.2×106
1000
–
Appendix
Table 6: Guidance-based sampling hyperparameters used for each PDE benchmark. All benchmarks use the same noise schedule parameters (σmin,σmax,ρ) and the same number of sampling iterations. Observation and PDE guidance are applied in a stage-wise manner, while Darcy additionally employs boundary guidance with separate step sizes.
Figure 3: Additional visualization illustrating reconstruction performance for Burgers’ equation. Each row corresponds to a different test sample. Columns show the sparse HF observations used for conditioning (obs.), the LF input, reconstructions obtained under different reconstruction regimes (R1–R3), and the HF reference solution (ref.). Relative reconstruction errors are reported below each reconstructed field.
Figure 4: Additional visualization illustrating reconstruction performance for Darcy flow. Each row corresponds to a different test sample. Columns show the sparse HF observations used for conditioning (obs.), the LF input, reconstructions obtained under different reconstruction regimes (R1–R3), and the HF reference solution (ref.). Relative reconstruction errors are reported below each reconstructed field.
Figure 5: Additional visualization illustrating reconstruction performance for Kolmogorov flow at frame 1. Each row corresponds to a different test sample. Columns show the sparse HF observations used for conditioning (obs.), the LF input, reconstructions obtained under different reconstruction regimes (R1–R3), and the HF reference solution (ref.). Relative reconstruction errors are reported below each reconstructed field.
Figure 6: Additional visualization illustrating reconstruction performance for Kolmogorov flow at frame 20. Each row corresponds to a different test sample. Columns show the sparse HF observations used for conditioning (obs.), the LF input constructed from sparse observations via interpolation, reconstructions obtained under different reconstruction regimes (R1–R3), and the HF reference solution (ref.). Relative reconstruction errors are reported below each reconstructed field.
Figure 7: Additional visualization illustrating reconstruction performance for Kolmogorov flow at frame 38. Each row corresponds to a different test sample. Columns show the sparse HF observations used for conditioning (obs.), the LF input constructed from sparse observations via interpolation, reconstructions obtained under different reconstruction regimes (R1–R3), and the HF reference solution (ref.). Relative reconstruction errors are reported below each reconstructed field.
Figure 8: Inference dynamics comparison between PICSB and guidance-based sampling for Burgers’ equation.
Figure 9: Inference dynamics comparison between PICSB and guidance-based sampling for Darcy flow.
Figure 10: Inference dynamics comparison between PICSB and guidance-based sampling for Kolmogorov flow at frame 20.
Figure 11: Reconstruction results under 5% sparse observations for the Burgers’ equation.
Figure 12: Reconstruction results under 1% sparse observations for the Burgers’ equation.
Figure 13: Inference under noisy observations for Burgers’ equation with randomly resampled sensors (R1). Each row corresponds to a different test sample, and each column corresponds to a noise level, with the HF reference solution shown in the rightmost column. Relative reconstruction errors are reported below each predicted field.
Figure 14: Inference under noisy observations for Burgers’ equation with per-sample fixed sensors (R2). Each row corresponds to a different test sample, and each column corresponds to a noise level, with the HF reference solution shown in the rightmost column. Relative reconstruction errors are reported below each predicted field.
Figure 15: Inference under noisy observations for Burgers’ equation with a fixed sensor set (R3). Each row corresponds to a different test sample, and each column corresponds to a noise level, with the HF reference solution shown in the rightmost column. Relative reconstruction errors are reported below each predicted field.
Figure 16: Inference under noisy observations for Darcy flow with randomly resampled sensors (R1). Each row corresponds to a different test sample, and each column corresponds to a noise level, with the HF reference solution shown in the rightmost column. Relative reconstruction errors are reported below each predicted field.
Figure 17: Inference under noisy observations for Darcy flow with a fixed sensor set (R3). Each row corresponds to a different test sample, and each column corresponds to a noise level, with the HF reference solution shown in the rightmost column. Relative reconstruction errors are reported below each predicted field.
Figure 18: Inference under noisy sparse observations for Kolmogorov flow with randomly resampled sensors (R1). Each row corresponds to a different test sample, and each column corresponds to a noise level, with the HF reference solution shown in the rightmost column. Relative reconstruction errors are reported below each predicted field.
Figure 19: Inference under noisy observations for Kolmogorov flow with per-sample fixed sensors (R2). Each row corresponds to a different test sample, and each column corresponds to a noise level, with the HF reference solution shown in the rightmost column. Relative reconstruction errors are reported below each predicted field.
Figure 20: Inference under noisy observations for Kolmogorov flow with a fixed sensor set (R3). Each row corresponds to a different test sample, and each column corresponds to a noise level, with the HF reference solution shown in the rightmost column. Relative reconstruction errors are reported below each predicted field.
Figure 21: Inference under noisy observations for Kolmogorov flow with a fixed sensor set (R3). Each row corresponds to a different test sample, and each column corresponds to a noise level, with the HF reference solution shown in the rightmost column. Relative reconstruction errors are reported below each predicted field.
Figure 22: Finetuning and inference results under 1% sparse observations for the Burgers’ equation with integrated finetuning.
Figure 23: Finetuning and inference results under 1% sparse observations for Darcy flow with integrated finetuning.
Figure 24: Finetuning and inference results under 1% sparse observations for the Kolmogorov flow with integrated finetuning.
Figure 25: Finetuning and inference results under 5% sparse observations for the Burgers’ equation with integrated finetuning.
Figure 26: Finetuning and inference results under 5% sparse observations for Darcy flow with integrated finetuning.
Figure 27: Finetuning and inference results under 5% sparse observations for the Kolmogorov flow with integrated finetuning.
Partial differential equation (PDE) surrogate modeling often relies on either dense observations or access to governing equations/physics constraints. In practice, observations can be sparse and partial, and the underlying physics may be partially specified or unavailable. We introduce SBNO, an equation-free paired-data diffusion bridge matching framework that learns conditional operators for both forward and inverse PDE problems under partial observations. SBNO uses diffusion Schrödinger bridge matching to learn a pair of time-dependent drifts from a mask-aware endpoint coupling that preserves empirical input-solution pairing during training and supports both PDE-task directions within a single framework. At inference time, SBNO produces a deterministic point reconstruction by integrating the probability-flow ODE constructed from the learned drift pair, without inference-time observation or PDE guidance-weight tuning. On five PDE benchmarks with 5% observations, SBNO achieves superior or comparable accuracy to baselines while demonstrating lower test-instance error dispersion. Furthermore, SBNO is at least 9.7× faster and reduces GPU memory by at least 4.3× compared to guided-diffusion baselines.
Generating and predicting spatiotemporal physical fields from scarce measurements is challenging, as observations are insufficient to characterize a distribution over complete fields. This limits conventional data-driven diffusion models that rely on full-field datasets. We introduce PhysDEM, a physics-defined diffusion framework that combines governing equations with spatially sparse observations to generate multiple plausible fields. First, we construct a Gibbs target by reweighting a measurement-conditioned Gaussian reference with PDE residual energy. Second, we derive an exact conditional-mean identity that reduces denoising to supervised learning of the standardized energy-induced mean correction. Third, a physics-displacement probability flow cancels Gaussian reference terms and enables amortized sampling with changing measurements through Gaussian conditioning, without retraining. Experiments on synthetic PDE systems and real-world-informed applications demonstrate that PhysDEM supports coherent field recovery and efficient sampling while maintaining stable diagnostics under tested noise levels, illustrating its practical value for field assessment. To our knowledge, PhysDEM is the first physics-defined diffusion model enabling amortized spatiotemporal field inference without preassembled full-field datasets.
Recovering complete physical fields from sparse observations is challenging because the measurements may not uniquely determine the underlying state. Diffusion-based PDE solvers address this problem through iterative sampling whereas neural operators provide deterministic one-pass predictions. We propose SCOPE (Sparse-Context Observability-aware Predictive Embeddings) to recover complete PDE fields from sparse observations by coupling full-field latent prediction with physical reconstruction. A shared decoder reconstructs fields from both predicted and complete-view representations so that representation learning is guided by both physical recovery and latent matching. We derive a quadratic risk decomposition at fixed teacher-decoder pairs showing why optimal latent prediction need not yield optimal field reconstruction. We also establish sufficient conditions for decoder improvements on complete inputs to transfer to recovery from partial observations. Experiments across five PDE settings show that SCOPE outperforms mask-aware neural operators on all ten forward and inverse tasks and achieves lower errors than those reported for diffusion-based solvers including DiffusionPDE and FunDPS. Decoder-only adaptation further improves recovery without retraining the backbone while retaining deterministic single-pass inference.
Ruichen Xu, Siyao Wang, Fang Wan +8
Stony Brook University · University of California, Davis · Independent Research +3