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
SBNO, an equation-free paired-data diffusion bridge matching framework, is introduced.
This framework learns conditional operators for both forward and inverse PDE problems under partial observations.
SBNO's Approach
SBN0 uses diffusion Schrödinger bridge matching to learn a pair of time-dependent drifts from a mask-aware endpoint coupling.
This preserves empirical input-solution pairing during training and supports both PDE-task directions within a single framework.
Inference Time
At inference time, SBNO produces a deterministic point reconstruction.
This is done by integrating the probability-flow ODE constructed from the learned drift pair.
This process does not require inference-time observation or PDE guidance-weight tuning.
Results
On five PDE benchmarks with 5% observations, SBNO achieves superior or comparable accuracy to baselines.
It also demonstrates lower test-instance error dispersion.
Performance Comparison
SBNO is at least $9.7 \times$ faster than guided-diffusion baselines.
It also reduces GPU memory by at least $4.3 \times$ compared to guided-diffusion baselines.