  We consider the problem of optimizing initial conditions and timing in
dynamical systems governed by unknown ordinary differential equations (ODEs),
where evaluating different initial conditions is costly and there are
constraints on observation times. To identify the optimal conditions within
several trials, we introduce a few-shot Bayesian Optimization (BO) framework
based on the system's prior information. At the core of our approach is the
System-Aware Neural ODE Processes (SANODEP), an extension of Neural ODE
Processes (NODEP) designed to meta-learn ODE systems from multiple trajectories
using a novel context embedding block. Additionally, we propose a
multi-scenario loss function specifically for optimization purposes. Our
two-stage BO framework effectively incorporates search space constraints,
enabling efficient optimization of both initial conditions and observation
timings. We conduct extensive experiments showcasing SANODEP's potential for
few-shot BO. We also explore SANODEP's adaptability to varying levels of prior
information, highlighting the trade-off between prior flexibility and model
fitting accuracy.
