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

• URL: http://arxiv.org/abs/2512.05297v1
• Date: Thu, 04 Dec 2025 22:33:29 GMT
• Title: CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators
• Authors: Xianglong Hou, Xinquan Huang, Paris Perdikaris,
• Abstract summary: Continuous Flow Operator (CFO) learns continuous-time PDE dynamics without the computational burden of standard continuous approaches, e.g., neural ODE.<n>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.<n>A neural operator is then trained via flow matching to predict these analytic velocity fields.
• Score: 9.273461312644345
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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 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.

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