Research map / Probabilistic and scientific modeling / Differential equations and PDEs

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

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural operators for general geometries

Read the paperCheck your idea against 108,967 papers

AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.

Proposes SBNO, an equation-free, mask-aware diffusion Schrödinger bridge matching neural operator that learns a unified forward/inverse drift pair from paired PDE data and performs deterministic ODE-based reconstruction under 5% observations.

Problem

Surrogate modeling of PDE forward and inverse problems when observations are sparse and partial and governing equations are unavailable or only partially specified.

Approach

Fixed empirical endpoint coupling of mask-lifted input-solution pairs (partial input to full solution and full input to partial solution) is used to train a pair of time-dependent drifts via diffusion Schrödinger bridge matching (DSBM); inference integrates the probability-flow ODE deterministically with no guidance-weight tuning.

Key results

• SBNO achieves the best or second-best mean relative L2 error in all ten forward/inverse settings on five benchmarks at 5% observations (e.g., Darcy forward 0.0359 vs FunDPS 0.0461).
• Average error dispersion reduced by 44.91%, from 0.0416 (FunDPS) to 0.0229 (SBNO).
• SBNO is at least 9.7x faster per sample (0.28 s vs 2.71 s for FunDPS) and uses at least 4.3x less peak GPU memory (886 MB vs 3,846.7 MB for DiffusionPDE) than guided diffusion baselines.

Surprising findings

• Deterministic ODE inference is not guaranteed to preserve the empirical endpoint coupling or recover the conditional mean, and cannot represent multimodal uncertainty.
• The authors could not reproduce FunDPS's originally reported accuracy and excluded the bounded Navier-Stokes dataset due to data-quality anomalies in the released version.

Limitations

• Assumes discretized fields on a regular grid; off-grid sensor observations require projection or interpolation.
• Deterministic ODE gives a single point reconstruction, cannot represent multimodal uncertainty or recover conditional mean.
• Requires fully observed input-solution pairs during training.

Experimental setup

Datasets: DiffusionPDE benchmark (Darcy flow, Poisson, Helmholtz, Nonbounded Navier-Stokes) The Well (Gray-Scott) APEBench (KdV, Kuramoto-Sivashinsky, Kolmogorov flow; appendix)

Benchmarks: DiffusionPDE Darcy flow DiffusionPDE Poisson DiffusionPDE Helmholtz DiffusionPDE Nonbounded Navier-Stokes The Well Gray-Scott

Metrics: relative L2 error mean relative L2 error std across test samples wall-clock time per sample peak GPU memory

Baselines: DiffusionPDE FunDPS FNO GNOT FNOPE

Builds on: diffusion Schrödinger bridge matching (DSBM) Schrödinger bridge probability-flow ODE Fourier Neural Operator

Closest papers by meaning

• Sparsely-Supervised Data Assimilation via Physics-Informed Schrödinger Bridge (NeurIPS 2026 · similarity 0.90)
• Feedback Schrödinger Bridge Matching (ICLR 2025 · similarity 0.88)
• Guided Diffusion Sampling on Function Spaces with Applications to PDEs (NeurIPS 2025 · similarity 0.88)
• Generalized Schrödinger Bridge Matching (ICLR 2024 · similarity 0.88)
• Momentum Multi-Marginal Schrödinger Bridge Matching (NeurIPS 2025 · similarity 0.88)
• Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems (ICLR 2025 · similarity 0.87)