Residual-augmented flow matching operators for probabilistic partial differential equations
stat.CO, cs.LG, stat.ML
Submitted: 2025-12-14
Updated: 2026-09-05
Code: https://github.com/sahilbhola14/floral
License: http://creativecommons.org/licenses/by/4.0/
The gist: Learning surrogate models for physical systems with latent uncertainty remains challenging in data-scarce regimes: deterministic neural operators fail to characterize uncertainty, while generative
Terminology
Abstract
Learning surrogate models for physical systems with latent uncertainty remains challenging in data-scarce regimes: deterministic neural operators fail to characterize uncertainty, while generative approaches require large ensembles of high-fidelity solution operator simulations and often sacrifice resolution generalizability. In this work, we propose a residual-augmented probabilistic operator learning framework that casts flow-matching-based generative modeling in infinite-dimensional function spaces while leveraging inexpensive low-fidelity solution operators as an inductive bias. Rather than learning the full high-fidelity stochastic solution operator directly, the proposed framework learns probabilistic residual operators that characterize the discrepancy between low- and high-fidelity solutions. By parameterizing the vector field in flow matching using neural operators conditioned on both the known system input and low-fidelity solution, the framework amortizes probabilistic inference across input conditions while enabling uncertainty-aware and resolution-generalizable predictions across spatial discretizations. Numerical experiments on stochastic advection, Burgers', and Darcy flow systems demonstrate that the residual-augmented formulation improves predictive accuracy under the same high-fidelity data budget, while the probabilistic operator learning formulation enables accurate characterization of uncertainty in low-data regimes compared to learning high-fidelity stochastic operators directly from data.
Sources
- Stochastic Flow Matching for Resolving Small-Scale Physics
- CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators
- Diffusion Generative Models in Infinite Dimensions
- Functional Flow Matching
- A Library for Learning Neural Operators
- Fourier Neural Operator for Parametric Partial Differential Equations
- Neural Operator: Graph Kernel Network for Partial Differential Equations
- Flow Matching for Generative Modeling
- Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs
- E-PINNs: Epistemic Physics-Informed Neural Networks
- Infinite-Dimensional Diffusion Models
- Generative Adversarial Neural Operators
- Stochastic Process Learning via Operator Flow Matching
- Denoising Diffusion Implicit Models
- Multi-fidelity wavelet neural operator with application to uncertainty quantification
- Improving and generalizing flow-based generative models with minibatch optimal transport
- A Denoising Diffusion Model for Fluid Field Prediction
- Guided Diffusion Sampling on Function Spaces with Applications to PDEs
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