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

Xianglong Hou, Xinquan Huang, Paris Perdikaris

TL;DR

<3-5 sentence high-level summary> CFO introduces a continuous-time neural operator for learning PDE dynamics by matching the analytic velocity of spline-based probability paths to a neural RHS. By reframing training through flow matching and using high-order splines, CFO trains on irregular time grids and supports inference at arbitrary temporal resolutions, avoiding backpropagation through ODE solvers. Across four benchmarks, CFO with only 25% irregular data outperforms autoregressive baselines trained on full data, demonstrating strong long-horizon stability and data efficiency. The framework is architecture-agnostic, supports reverse-time inference, and offers time-resolution invariance, enabling practical neural PDE surrogates for irregularly sampled real-world data.

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.

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

TL;DR

<3-5 sentence high-level summary> CFO introduces a continuous-time neural operator for learning PDE dynamics by matching the analytic velocity of spline-based probability paths to a neural RHS. By reframing training through flow matching and using high-order splines, CFO trains on irregular time grids and supports inference at arbitrary temporal resolutions, avoiding backpropagation through ODE solvers. Across four benchmarks, CFO with only 25% irregular data outperforms autoregressive baselines trained on full data, demonstrating strong long-horizon stability and data efficiency. The framework is architecture-agnostic, supports reverse-time inference, and offers time-resolution invariance, enabling practical neural PDE surrogates for irregularly sampled real-world data.

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.

Paper Structure

This paper contains 88 sections, 2 theorems, 45 equations, 12 figures, 14 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Neural PDE Solvers and Operator Learning.
  4. Neural ODEs for Continuous-time Modeling.
  5. Generative Models for PDE Simulation.
  6. Our Approach.
  7. Preliminaries
  8. Method of Lines.
  9. Stochastic Interpolants and Flow Matching.
  10. Continuous Flow Operator
  11. Methodology
  12. Notation.
  13. Problem Statement.
  14. Probability Path Formulation.
  15. Finite Difference Approximation.
  16. ...and 73 more sections

Key Result

Proposition A.1

(Extension of Theorem 2.6 in albergo2023stochastic2) Under the definitions and Assumption ass:grouped, the random variable $I(t;\mathbf u) = s(t;\mathbf u)+\gamma(t)Z$ admits a strictly positive (Lebesgue) density $\rho(t,\cdot)$ for every $t\in[0,1]$. Moreover:

Figures (12)

  • Figure 1: Overview of the Continuous Flow Operator (CFO) framework. (a) Training: flow matching on a spline path. For each trajectory with snapshots $\{u(t_i)\}$, we fit a temporal spline $s(t)$ that interpolates the data and matches finite-difference derivative estimates at knots. The spline's analytic time derivative $\partial_t s(t)$ provides exact velocity targets for training a neural operator $\mathcal{N}_\theta$ via the flow matching objective—no ODE integration required during training. (b) Inference: continuous-time rollout. The trained operator $\mathcal{N}_\theta(t,u)$ defines a continuous vector field $\dot{u}_\theta = \mathcal{N}_\theta(t,u_\theta)$. Given initial condition $u(t)$, we compute $u(t')$ by numerical integration; reverse-time inference integrates backward.
  • Figure 2: Spline fitting error on the Lorenz system. Average velocity-field approximation error for quintic and linear splines at different time-step sizes ($\Delta t$, $2\Delta t$, $4\Delta t$), with $\Delta t = 0.005\text{s}$.
  • Figure 3: Training comparison between linear and quintic CFO on the Lorenz system. Training loss (10k–60k iterations) for linear and quintic CFO under different keep ratios: (a) 100%, (b) 50%, (c) 25%.
  • Figure 4: Worst-case prediction for Burgers' equation. Quintic CFO (trained on 25% irregular data) accurately predicts the full trajectory for the test sample with the highest error. Shown from left to right: solutions at the 25, 50, 75, and 100th time steps.
  • Figure 5: DR trajectory visualization for activator (top) and inhibitor (bottom).
  • ...and 7 more figures

Theorems & Definitions (4)

  • Proposition A.1
  • proof
  • Proposition A.2
  • proof