Learning effective stochastic differential equations from microscopic simulations: Linking stochastic numerics to deep learning.

We identify effective stochastic differential equations (SDEs) for coarse observables of fine-grained particle- or agent-based simulations; these SDEs then provide useful coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDEs through neural networks, which can be thought of as effective stochastic ResNets. The loss function is inspired by, and embodies, the structure of established stochastic numerical integrators (here, Euler-Maruyama and Milstein); our approximations can thus benefit from backward error analysis of these underlying numerical schemes. They also lend themselves naturally to "physics-informed" gray-box identification when approximate coarse models, such as mean field equations, are available. Existing numerical integration schemes for Langevin-type equations and for stochastic partial differential equations can also be used for training; we demonstrate this on a stochastically forced oscillator and the stochastic wave equation. Our approach does not require long trajectories, works on scattered snapshot data, and is designed to naturally handle different time steps per snapshot. We consider both the case where the coarse collective observables are known in advance, as well as the case where they must be found in a data-driven manner.

MarkerMap: nonlinear marker selection for single-cell studies.

Single-cell RNA-seq data allow the quantification of cell type differences across a growing set of biological contexts. However, pinpointing a small subset of genomic features explaining this variability can be ill-defined and computationally intractable. Here we introduce MarkerMap, a generative model for selecting minimal gene sets which are maximally informative of cell type origin and enable whole transcriptome reconstruction. MarkerMap provides a scalable framework for both supervised marker selection, aimed at identifying specific cell type populations, and unsupervised marker selection, aimed at gene expression imputation and reconstruction. We benchmark MarkerMap's competitive performance against previously published approaches on real single cell gene expression data sets. MarkerMap is available as a pip installable package, as a community resource aimed at developing explainable machine learning techniques for enhancing interpretability in single-cell studies.

On the parameter combinations that matter and on those that do not: data-driven studies of parameter (non)identifiability.

We present a data-driven approach to characterizing nonidentifiability of a model's parameters and illustrate it through dynamic as well as steady kinetic models. By employing Diffusion Maps and their extensions, we discover the minimal combinations of parameters required to characterize the output behavior of a chemical system: a set of effective parameters for the model. Furthermore, we introduce and use a Conformal Autoencoder Neural Network technique, as well as a kernel-based Jointly Smooth Function technique, to disentangle the redundant parameter combinations that do not affect the output behavior from the ones that do. We discuss the interpretability of our data-driven effective parameters, and demonstrate the utility of the approach both for behavior prediction and parameter estimation. In the latter task, it becomes important to describe level sets in parameter space that are consistent with a particular output behavior. We validate our approach on a model of multisite phosphorylation, where a reduced set of effective parameters (nonlinear combinations of the physical ones) has previously been established analytically.