Now, we’re going back to basics.
Assume we have a physical model
∂tu=f(ut−1,t;θ) We can solve this using classical PDEs
ut=ODESolve(f,u0,t,θ) Data¶
D={yt}t=1Tyt∈RDy Model¶
Initial:Dynamical Model:Observation Model:u0utyt∼N(u0∣ub,Σb)=ODESolve(f,ut−1,t−1,θ)=h(ut,t;θ)+εy Criteria¶
L(θ)=t=1∑T∣∣yt−h(ut,t;θ)∣∣Σy−12+∣∣u0−ub∣∣Σb−12 Here, the parameters are
θ={θ,ub,Σb} Latent Dynamical Model¶
Model¶
Initial Latent Dist.:Latent Dynamical Model:Observation Model:z0ztyt∼p(z0∣θ)=f(zt−1,t;θ)+εz=h(zt,t;θ)+εy Criteria¶
L(θ,ϕ;D)=t=1∑TEqϕ(zt−1)[logpθ(zt∣zt−1)+logpθ(yt∣zt)−logqϕ(zt∣zt−1)] where
Prior:p(zt∣zt−1;θ)=N(zt∣fθe(zt−1);)