Bridging the Physics-Data Gap with FNO-Guided Conditional Flow Matching: Designing Inductive Bias through Hierarchical Physical Constraints

Conventional time-series generation often ignores domain-specific physical constraints, limiting statistical and physical consistency. We propose a hierarchical framework that embeds the inherent hierarchy of physical laws-conservation, dynamics, boundary, and empirical relations-directly into deep generative models, introducing a new paradigm of physics-informed inductive bias. Our method combines Fourier Neural Operators (FNOs) for learning physical operators with Conditional Flow Matching (CFM) for probabilistic generation, integrated via time-dependent hierarchical constraints and FNO-guided corrections. Experiments on harmonic oscillators, human activity recognition, and lithium-ion battery degradation show 16.3% higher generation quality, 46% fewer physics violations, and 18.5% improved predictive accuracy over baselines.