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Sparsely-Supervised Data Assimilation via Physics-Informed Schrödinger Bridge

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Physics-informed operator learning · 0 citations (OpenAlex)

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Proposes PICSB, a guidance-free conditional Schrödinger bridge that maps an LF field and sparse HF observations to HF reconstructions, trained only with PDE residuals via a surrogate-endpoint refresh and hard observation projection.

Problem

Amortized multi-fidelity data assimilation for PDE systems: reconstructing full HF spatiotemporal fields from sparse HF observations and an LF field, without per-instance optimization and without full-field HF supervision.

Approach

Conditional Schrödinger bridge transporting an LF-informed prior toward an observation-conditioned HF posterior, trained solely by physics residuals, with hard observation projection at every sampling step and a periodically refreshed surrogate terminal reference replacing unavailable HF endpoints.

Key results

• On Darcy, PICSB achieves 0.29% relative error under R1 versus 5.69% for PINNs and 12.02% for Guidance.
• On Kolmogorov under R1, PICSB reaches 7.32% error versus 8.99% for Guidance and 64.47% for PINNs.
• Inference wall-time is 0.023 s on Burgers and Darcy and 0.134 s on Kolmogorov, versus 173-801 s for Guidance and 1733-10680 s for PINNs.

Surprising findings

• On Burgers under R1, Guidance (0.99%) beats PICSB (2.59%), and PICSB is not best under R2 and R3 Burgers settings.
• Under R2 and R3 Burgers, PINNs matches or beats PICSB (13.48% vs 14.05% in R2).

Limitations

• Assumes known PDE operator and known observation operator; extension to unknown or learned operators left to future work.
• Accuracy is not uniformly best; Burgers results are competitive but trail baselines in several regimes.
• Evaluated on only 4 test instances and 32 training pairs per benchmark, with small-scale benchmarks.

Experimental setup

Datasets: 1D viscous Burgers simulations (512x512 HF, 128x128 downsampled) 2D Darcy flow (128x128) 2D turbulent Kolmogorov flow (256x256, 40 frames)

Benchmarks: 1D Burgers 2D Darcy flow 2D Kolmogorov flow

Metrics: relative L2 error (%) wall-time (s)

Baselines: PINNs Guidance (PDE-guided diffusion with EDM sampler)

Builds on: Schrödinger bridge iterative Markovian fitting EDM diffusion sampler PINNs target network

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