# CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators

arXiv: https://arxiv.org/abs/2512.05297 · HTML: https://mlsanity.com/paper/2512.05297/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 29 of its 35 references that are in the map.
- First posted on arXiv 4 Dec 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 29 of 29)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (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)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems](https://mlsanity.com/paper/2312.10527/) (2023)
- [CViT: Continuous Vision Transformer for Operator Learning](https://mlsanity.com/paper/2405.13998/) (2024)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [CurveFlow: Curvature-Guided Flow Matching for Image Generation](https://mlsanity.com/paper/2508.15093/) (2025)
- [Reduced-order modeling for parameterized PDEs via implicit neural representations](https://mlsanity.com/paper/2311.16410/) (2023)

## Built on by (most cited first, 11 of 11)

- [FlowRefiner: Flow Matching-Based Iterative Refinement for 3D Turbulent Flow Simulation](https://mlsanity.com/paper/2604.17149/) (2026)
- [Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing](https://mlsanity.com/paper/2604.07366/) (2026)
- [Residual-augmented flow matching operators for probabilistic partial differential equations](https://mlsanity.com/paper/2512.12749/) (2025)
- [Recovering Physical Dynamics from Discrete Observations via Intrinsic Differential Consistency](https://mlsanity.com/paper/2605.08454/) (2026)
- [Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow](https://mlsanity.com/paper/2605.08915/) (2026)
- [Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions](https://mlsanity.com/paper/2605.16118/) (2026)
- [Gradient-Informed Temporal Sampling Improves Rollout Accuracy in PDE Surrogate Training](https://mlsanity.com/paper/2603.18237/) (2026)
- [Approximation of Maximally Monotone Operators : A Graph Convergence Perspective](https://mlsanity.com/paper/2605.12301/) (2026)
- [Sparse Compositional Flow Matching by geometric assembly from motion primitives](https://mlsanity.com/paper/2605.23341/) (2026)
- [Neural Constitutive Learning for Generalized Reaction-Diffusion Systems](https://mlsanity.com/paper/2609.37113/) (2026)
- [Physics is the Best Teacher: Consistency Learning for Time-Invariant Operators of Chaotic Dynamics](https://mlsanity.com/paper/2610.04108/) (2026)

## Abstract (arXiv)

Neural operator surrogates for time-dependent partial differential equations (PDEs) conventionally employ autoregressive prediction schemes, which accumulate error over long rollouts and require uniform temporal discretization. We introduce the Continuous Flow Operator (CFO), a framework that learns continuous-time PDE dynamics without the computational burden of standard continuous approaches, e.g., neural ODE. The key insight is repurposing flow matching to directly learn the right-hand side of PDEs without backpropagating through ODE solvers. CFO fits temporal splines to trajectory data, using finite-difference estimates of time derivatives at knots to construct probability paths whose velocities closely approximate the true PDE dynamics. A neural operator is then trained via flow matching to predict these analytic velocity fields. This approach is inherently time-resolution invariant: training accepts trajectories sampled on arbitrary, non-uniform time grids while inference queries solutions at any temporal resolution through ODE integration. Across four benchmarks (Lorenz, 1D Burgers, 2D diffusion-reaction, 2D shallow water), CFO demonstrates superior long-horizon stability and remarkable data efficiency. CFO trained on only 25% of irregularly subsampled time points outperforms autoregressive baselines trained on complete data, with relative error reductions up to 87%. Despite requiring numerical integration at inference, CFO achieves competitive efficiency, outperforming autoregressive baselines using only 50% of their function evaluations, while uniquely enabling reverse-time inference and arbitrary temporal querying.