SBNO : Schrödinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information

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 learns a pair of time-dependent forward/backward drifts via diffusion Schrödinger bridge matching, using a unified endpoint coupling that preserves empirical input-solution pairing, enabling both directions to be handled within a single framework. At inference time, SBNO performs deterministic reconstruction by integrating the probability-flow ODE of the learned bridge, eliminating the need for inference-time hyperparameter 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 $9.7\times$ faster and reduces GPU memory by at least $4.3\times$ compared to diffusion-based baselines.