A Mechanistic Dynamic Emulator

In applied sciences, we often deal with deterministic simulation models that are too slow for simulation-intensive tasks such as calibration or real-time control. In this paper, an emulator for a generic dynamic model, given by a system of ordinary non-linear differential equations, is developed. The non-linear differential equations are linearized and Gaussian white noise is added to account for the non-linearities. The resulting linear stochastic system is conditioned on a set of solutions of the non-linear equations that have been calculated prior to the emulation. A path-integral approach is used to derive the Gaussian distribution of the emulated solution. The solution reveals that most of the computational burden can be shifted to the conditioning phase of the emulator and the complexity of the actual emulation step only scales like $\mathcal O(Nn)$ in multiplications of matrices of the dimension of the state space. Here, $N$ is the number of time-points at which the solution is to be emulated and $n$ the number of solutions the emulator is conditioned on. The applicability of the algorithm is demonstrated with the hydrological model logSPM.