New Linear Response Methods for Data Assimilation and Generative Modeling

We present two new formulas for linear response (parameter
derivatives of marginal or stationary measures) of random
dynamical systems. These formulas unify classical
approaches, including path-perturbation, divergence, and
kernel-differentiation methods, while overcoming major
difficulties such as chaos, high dimensionality, and
parameter-dependent multiplicative noise.

Using the new adjoint path-kernel formula, we solve the data
assimilation problem in a general setting with chaotic
high-dimensional dynamics, partial and noisy observations,
unknown parameters, a single loss over a long time window,
and no synchronization assumption.

Using the new divergence-kernel formula, we introduce a
generative modeling framework, DK-SDE, in which the model is
a parameterized nonlinear SDE trained by minimizing the KL
divergence between the data distribution and the SDE
marginal law. This framework naturally incorporates explicit
dynamical priors and has substantially lower memory cost.