Energy-Guided Functional Flow Matching for Soliton Generation

We study energy-guided Functional Flow Matching for generating solutions of the Korteweg–de Vries equation. A Fourier Neural Operator parameterizes a velocity field that transports a Gaussian reference measure toward a distribution of soliton solutions. We consider both complete space–time trajectories and joint initial–final state representations. Under explicit measure-theoretic assumptions, we derive an exact guidance identity in function space and a tractable covariance-based approximation for sampling. On the reported two- and three-soliton forecasting benchmarks, FFM with initial-condition guidance achieves lower median trajectory errors and PDE residuals than our DiffusionPDE and FunDPS implementations. Additional invariant guidance reduces momentum and energy drift, while its effect on prediction error depends on the soliton regime. These results distinguish conservation from dynamical accuracy and support guided functional generation as a promising approach to dispersive PDEs. The experiments assess performance on a common grid.