CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators
Xianglong Hou, Xinquan Huang, Paris Perdikaris
TL;DR
CFO introduces a continuous-time PDE learning framework using flow matching, enabling stable long-term predictions, arbitrary temporal resolution, and high data efficiency without the need for backpropagating through ODE solvers.
Contribution
It presents a novel flow-matching based method for learning PDE dynamics directly in continuous time, avoiding traditional autoregressive schemes and uniform discretization constraints.
Findings
Outperforms autoregressive baselines in long-horizon stability.
Requires only 25% of data points to achieve superior accuracy.
Enables efficient inference at arbitrary temporal resolutions.
Abstract
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…
Peer Reviews
Decision·ICLR 2026 Poster
The main strengths of the paper are as follows: - It proposes an approach that significantly reduces the computational and memory costs associated with applying directly neural ODE frameworks to PDE prediction tasks. - It bridges flow learning and stochastic modeling by introducing a strategy to estimate temporal derivatives at unobserved time points through stochastic interpolation of training snapshots.
- Limited baselines: The paper only compares against a discrete-time autoregressive model. While neural ODE-based approaches can indeed be computationally demanding, this alone is not a sufficient reason to omit them from quantitative comparison. Including related continuous-time baselines such as the mentioned continuous-time methods (e.g., Chen et al., 2018 for ODE/PDEs; Yin et al., 2023 for PDEs) or other comparable approaches would strengthen the experimental validation. - Unclear training
From an originality perspective, the idea of repurposing flow matching, which is originally developed for CNFs/diffusion, to directly learn RHS dynamics is elegant and avoids the heavy simulation-through-solvers that slows Neural ODE/SDE training. The positioning is credible relative to Flow Matching in generative modeling and stochastic interpolants, as CFO leverages fixed probability paths defined by the spline while keeping the operator-learning focus. I appreciate that the authors discuss fl
The reverse-time inference claim is attractive, but dissipative PDEs often make backward integration ill-posed. I would prefer a short quantitative check showing how error grows with backward horizon and perturbation size to avoid over-selling. Baseline coverage against continuous-time sequence models could be expanded. Since CFO’s training avoids solver backprop similarly to newer flow-matching approaches for time series, a comparison to Trajectory Flow Matching (TFM) would situate the contrib
1. This paper is well-organized and highly readable. 2. The adoption of the flow matching concept can significantly reduce the training complexity of neural networks, and may even enhance the learning accuracy in certain scenarios. 3. The author has conducted experimental analyses across multiple datasets.
1. The work lacks originality, as the idea of applying flow matching to train Neural ODEs has been previously explored in several earlier studies. 2. The experimental section lacks validation on real-world datasets, which significantly limits the persuasiveness of the claims. 3. It lacks comparison with existing SOTA baseline methods.
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Taxonomy
TopicsModel Reduction and Neural Networks · Neural Networks and Reservoir Computing · Machine Learning in Materials Science