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m →Non-equilibrium thermodynamics: Word choice ("learn" meaning to teach is an archaic use of the word. The paper refers to what they do with models as "train") |
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==== Backward diffusion process ====
If we have solved <math>\rho_t</math> for time <math>t\in [0, T]</math>, then we can exactly reverse the evolution of the cloud. Suppose we start with another cloud of particles with density <math>\nu_0 = \rho_T</math>, and let the particles in the cloud evolve according to
<math display="block">dy_t = \frac{1}{2} \beta(T-t) y_{t} d t + \beta(T-t) \underbrace{\nabla_{y_{t}} \ln \rho_{T-t}\left(y_{t}\right)}_{\text {score function }} d t+\sqrt{\beta(T-t)} d W_t</math> then by plugging into the Fokker-Planck equation, we find that <math>\partial_t \rho_{T-t} = \partial_t \nu_t</math>. Thus this cloud of points is the original cloud, evolving backwards.<ref>{{Cite journal |last=Anderson |first=Brian D.O. |date=May 1982 |title=Reverse-time diffusion equation models |url=http://dx.doi.org/10.1016/0304-4149(82)90051-5 |journal=Stochastic Processes and Their Applications |volume=12 |issue=3 |pages=313–326 |doi=10.1016/0304-4149(82)90051-5 |issn=0304-4149}}</ref> === Noise conditional score network (NCSN) ===
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