# SBNO : Schr"odinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information

arXiv: https://arxiv.org/abs/2610.13317 · HTML: https://mlsanity.com/paper/2610.13317/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 40 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 27 of 27)

- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Denoising Diffusion Bridge Models](https://mlsanity.com/paper/2309.16948/) (2023)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs](https://mlsanity.com/paper/2411.00180/) (2024)
- [Generative Adversarial Neural Operators](https://mlsanity.com/paper/2205.03017/) (2022)
- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Stability of Entropic Optimal Transport and Schr"odinger Bridges](https://mlsanity.com/paper/2106.03670/) (2021)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems](https://mlsanity.com/paper/2412.03161/) (2024)
- [Operator learning meets inverse problems: A probabilistic perspective](https://mlsanity.com/paper/2508.20207/) (2025)
- [FUSE: Fast Unified Simulation and Estimation for PDEs](https://mlsanity.com/paper/2405.14558/) (2024)
- [FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators](https://mlsanity.com/paper/2505.22573/) (2025)

## Abstract (arXiv)

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 Schr"odinger 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 × faster and reduces GPU memory by at least 4.3 × compared to guided-diffusion baselines.