Flow Algorithm
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1. Overview
2. Type 1. CNF
3. Type 2. DOT
4. Type 3. Flow matching
1. Overview
⑴ Streamline: for vector field f(x, t) and position x(t)
Figure 1. Example of a streamline
⑵ Continuity equation: by the law of conservation of mass, the density ρ(x, t) must satisfy the following
⑶ Flow algorithm
⑷ The uniqueness of f becomes an issue, and if uniqueness is satisfied, it becomes the domain of causal inference (ref)
2. Type 1. CNF
⑴ Continuous normalizing flow (CNF): learning a complex distribution ρ(x, t1) from a simple distribution ρ(x, t0), as in a diffusion model (Grathwohl et al., 2019)
⑵ In actual papers, it is expressed more specifically as follows
3. Type 2. DOT
⑴ Optimal transport theory(optimal transport)
⑵ Dynamic optimal transport (DOT): constructs a continuous path that minimizes the cost of the trajectory connecting two time points t0 and t1. Such a cost is called the L2 Wasserstein distance
⑶ In the above equation, ρ ㅣfㅣ2 ≃ mv2 = 2K, which is closely related to energy.
⑷ Example 1. TrajectoryNet implements DOT using CNF.
⑸ Example 2. DeepRUOTv2
4. Type 3. Flow matching
⑴ Flow matching: although its objective is the same as CNF, it enables more scalable analysis through regression analysis (Lipman et al., 2023). For the estimated flow u(x, t),
⑵ If the regression analysis is perfect, then ρ = ρ’, so the continuity equation constraint is essentially no longer necessary.
⑶ Since ρ’(x, t) and u(x, t) are not given, the following equation is used:
Input: 2026.07.03 01:21