Divergence-Kernel method for linear responses and diffusion models
This work addresses the problem of efficiently computing linear responses in dynamical systems for researchers in applied mathematics and machine learning, though it appears incremental as it builds on existing methods.
The authors derived a divergence-kernel formula for computing linear responses in random dynamical systems, applicable to multiplicative and parameterized noise without requiring hyperbolicity, and proposed a forward-only diffusion generative model tested on simple problems.
We derive the divergence-kernel formula for the linear response (parameter-derivative of marginal or stationary distributions) of random dynamical systems, and formally pass to the continuous-time limit. Our formula works for multiplicative and parameterized noise over any period of time; it does not require hyperbolicity. Then we derive a pathwise Monte-Carlo algorithm for linear responses. With this, we propose a forward-only diffusion generative model and test on simple problems.