Schrödinger bridge method solves PDE problems with sparse data faster
SBNO : Schrödinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information
Machine Learning
Summary
Solving equations that describe physical systems usually needs a lot of data or the exact rules (physics) to follow. This paper introduces SBNO, a new way to figure out solutions even when there’s only a small amount of information and incomplete knowledge of the rules. SBNO learns to predict solutions forward and backward without being given the full equations, making it faster and using less computer memory. Tests show SBNO gives as good or better answers than older methods, even with only 5% of the data.
What this means in practice
- •For engineering simulation teams: Perform forward and inverse PDE simulations accurately with limited observational data and without full physics equations.
- •For weather forecasting modelers: Improve model accuracy using only sparse measurements when full physical climate models are unavailable or incomplete.
Authors
Jaehyeon Park, Inhyeok Jeong, Noseong Park
Abstract
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 $9.7 \times$ faster and reduces GPU memory by at least $4.3 \times$ compared to guided-diffusion baselines.