CMX Lunch Seminar

We present several new formulas for the linear response (parameter derivatives of marginal or stationary measures) of SDEs. The formulas subsume classical approaches (path-perturbation, divergence, and kernel-differentiation) and overcome key difficulties such as chaos, high-dimensionality, and parameterized noise.

With the new adjoint path-kernel formula, we solve a challenging variational data assimilation problem where (i) the deterministic dynamics is chaotic, (ii) the objective is a single long-time function measuring mismatch in both observations and dynamics, (iii) some dynamical parameters are unknown, and (iv) the state is only partially observed.

With another divergence-kernel formula, we introduce a generative model, DK-SDE, where the model is a parameterized SDE trained by minimizing the KL divergence between the data and the SDE marginal law. The framework allows parametrizations in both drift and diffusion (enabling explicit priors in dynamics), and its gradient computation uses only forward processes, substantially reducing memory cost.