{"ID":2833213,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.05297","arxiv_id":"2512.05297","title":"CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators","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.","short_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...","url_abs":"https://arxiv.org/abs/2512.05297","url_pdf":"https://arxiv.org/pdf/2512.05297v1","authors":"[\"Xianglong Hou\",\"Xinquan Huang\",\"Paris Perdikaris\"]","published":"2025-12-04T22:33:29Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.AI\",\"math.NA\"]","methods":"[\"Diffusion Model\"]","has_code":false}