# Normalizing Flow

KC-maps learns the finite-time transition kernel of a stochastic differential equation directly as a neural network map, trained only on uncorrelated marginal samples plus the known drift and diffusion coefficients rather than on paired simulated trajectories. Boundary conditions are built into the network architecture, and a training objective derived from the Kolmogorov–Chapman equation keeps transitions composed over different time scales consistent in distribution. On alanine dipeptide, tetrapeptide and hexapeptide it replaces up to 70 fs of sequential integration with a single learned transition, reaches lower TICA-2 distances at matched GPU-hour budgets, and on AL6 covers conformational modes absent from the training data.

KC-maps learns the finite-time transition kernel of a stochastic differential equation directly as a neural network map, trained only on uncorrelated marginal samples plus the known drift and diffusion coefficients rather than on paired simulated trajectories. Boundary conditions are built into the network architecture, and a training objective derived from the Kolmogorov–Chapman equation keeps transitions composed over different time scales consistent in distribution. On alanine dipeptide, tetrapeptide and hexapeptide it replaces up to 70 fs of sequential integration with a single learned transition, reaches lower TICA-2 distances at matched GPU-hour budgets, and on AL6 covers conformational modes absent from the training data.

In the dilepton channel, two neutrinos escape undetected, so reconstructing top-quark-pair kinematics has long relied on analytic approximations. Using a 7 TeV simulated sample from CMS Open Data, researchers first refined the loose kinematic reconstruction into LKRv3, then trained a residual multilayer perceptron and a conditional normalizing flow to regress the top-pair four-momenta directly from visible particles; relative to LKRv3, the two networks improve the resolution of the invariant mass, transverse momentum and rapidity by about 30%, 20% and 15%, reaching 30 GeV invariant-mass resolution near the production threshold.

In the dilepton channel, two neutrinos escape undetected, so reconstructing top-quark-pair kinematics has long relied on analytic approximations. Using a 7 TeV simulated sample from CMS Open Data, researchers first refined the loose kinematic reconstruction into LKRv3, then trained a residual multilayer perceptron and a conditional normalizing flow to regress the top-pair four-momenta directly from visible particles; relative to LKRv3, the two networks improve the resolution of the invariant mass, transverse momentum and rapidity by about 30%, 20% and 15%, reaching 30 GeV invariant-mass resolution near the production threshold.