The Cislo–Siggia (2025) landscape is built by discretizing a Fokker–Planck path integral on points sampled from the data, then reading fixed points and basins off the discrete operator

The method of claim-cislo-siggia-2025-fit-waddington-landscape-directly-to-single-cell-gene-expression treats cell state x(t) as governed by a Langevin equation — a gradient-like drift set by a potential U and a metric tensor g, plus noise — and solves the corresponding Fokker–Planck equation for how the probability distribution evolves. Rather than laying that equation on a grid (which scales exponentially with dimension), it restricts the dynamics to the data itself: "We propose to circumvent the exponential increase in computational resources with dimension by defining a Markov process restricted to representative points sampled from the data."

Concretely, "the discrete representation of our underlying dynamical manifold is a set of N points M̂ = {xi} sampled from the experimental data," drawn "from all time points and all experimental conditions in order to achieve robust coverage of all relevant regions of the gene expression space." A path-integral solution of the Fokker–Planck equation is discretized into a transition matrix on that point set.

The dynamical structure is then read directly off the discrete operator, with no prior reduction to a hypothesized low-dimensional geometry: "Our discrete operators allow us to directly infer the dynamical structure of a system, including fixed points, unstable manifolds, and basins of attraction, with minimal preprocessing." Fixed points, saddles, unstable manifolds, and basins are extracted using topological data analysis and the backward Kolmogorov equation.

Sampling "representative points" and running topology on them places this beside the vault's landmark/subsampling methodology thread — entity-landmark-selection-topological-data-analysis and claim-chazal-2014-persistence-diagram-subsampling-stable-under-noise-not-selection — though here the subsample defines a dynamics operator, not a persistence diagram. The Langevin/Fokker–Planck framing also rhymes with the vault's stochastic-dynamics cluster, e.g. claim-constant-sgd-near-a-loss-minimum-is-an-ornstein-uhlenbeck-process.