A cylindrical neural approximation theorem for conditional laws of McKean-Vlasov equations with common noise

What the math gives to ML

The paper provides a constructive representation for conditional laws driven by a shared stochastic path: encode the initial distribution with finitely many Fourier moments, encode the common-noise history with a truncated path signature, predict a Gaussian-mixture conditional law, and evaluate downstream functionals through a cylindrical network. The transferable asset is the separation of two sources of uncertainty: distributional state information and path-dependent shared context. Signature features offer a principled fixed-size representation of common-noise histories, while Gaussian mixtures provide an explicit differentiable approximation of the entire conditional distribution rather than only its moments. A practical neural implementation is a conditional-law module that replaces particle ensembles or ad hoc pooling with Fourier/signature features followed by a mixture-density head and differentiable expectation evaluation.