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

2025-12-04

AI Summary

Traditional autoregressive neural operators for time-dependent PDEs suffer from accumulated long-horizon errors and reliance on uniform temporal discretization. To address this, we propose the Continuous Flow Operator (CFO) framework—the first to incorporate flow matching for learning the PDE right-hand side, enabling direct modeling of continuous-time dynamics without backpropagation through ODE solvers. CFO synergistically integrates neural operators with trajectory spline fitting to estimate instantaneous time derivatives and construct a probability path approximating the true evolution, thereby training a robust velocity field. The method supports arbitrary-time queries, backward-in-time inference, and temporal resolution invariance. Evaluated on four benchmarks, CFO achieves up to 87% lower prediction error using only 25% irregularly sampled data, improves inference efficiency by 50%, and significantly enhances long-term forecasting accuracy and robustness.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Learning continuous-time PDE dynamics without autoregressive error accumulation
• Achieving time-resolution invariance for arbitrary non-uniform temporal sampling
• Enhancing long-horizon stability and data efficiency in PDE surrogate modeling

Innovation

• Uses flow matching to learn PDE dynamics directly
• Fits temporal splines to handle arbitrary time grids
• Enables resolution-invariant training and inference via ODE integration