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Register a new userLearning differential equation models from stochastic agent-based model simulations
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Agent-based models provide a flexible framework that is frequently used for modelling many biological systems, including cell migration, molecular dynamics, ecology, and epidemiology. Analysis of the model dynamics can be challenging due to their inherent stochasticity and heavy computational requirements. Common approaches to the analysis of agent-based models include extensive Monte Carlo simulation of the model or the derivation of coarse-grained differential equation models to predict the expected or averaged output from the agent-based model. Both of these approaches have limitations, however, as extensive computation of complex agent-based models may be infeasible, and coarse-grained differential equation models can fail to accurately describe model dynamics in certain parameter regimes. We propose that methods from the equation learning field provide a promising, novel, and unifying approach for agent-based model analysis. Equation learning is a recent field of research from data science that aims to infer differential equation models directly from data. We use this tutorial to review how methods from equation learning can be used to learn differential equation models from agent-based model simulations. We demonstrate that this framework is easy to use, requires few model simulations, and accurately predicts model dynamics in parameter regions where coarse-grained differential equation models fail to do so. We highlight these advantages through several case studies involving two agent-based models that are broadly applicable to biological phenomena: a birth-death-migration model commonly used to explore cell biology experiments and a susceptible-infected-recovered model of infectious disease spread.
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We identify effective stochastic differential equations (SDE) for coarse observables of fine-grained particle- or agent-based simulations; these SDE then provide coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDE 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 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. 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.
We investigate methods for learning partial differential equation (PDE) models from spatiotemporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyze the performance in utilizing previous methods to denoise data for the task of discovering the governing system of partial differential equations (PDEs). We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e., the advection-diffusion, classical Fisher-KPP, and nonlinear Fisher-KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model.
In a number of cases, the Quantile Gaussian Process (QGP) has proven effective in emulating stochastic, univariate computer model output (Plumlee and Tuo, 2014). In this paper, we develop an approach that uses this emulation approach within a Bayesian model calibration framework to calibrate an agent-based model of an epidemic. In addition, this approach is extended to handle the multivariate nature of the model output, which gives a time series of the count of infected individuals. The basic modeling approach is adapted from Higdon et al. (2008), using a basis representation to capture the multivariate model output. The approach is motivated with an example taken from the 2015 Ebola Challenge workshop which simulated an ebola epidemic to evaluate methodology.
In this note we prove that a fractional stochastic delay differential equation which satisfies natural regularity conditions generates a continuous random dynamical system on a subspace of a Holder space which is separable.
Since its inception, control of data congestion on the Internet has been based on stochastic models. One of the first such models was Random Early Detection. Later, this model was reformulated as a dynamical system, with the average queue sizes at a routers buffer being the states. Recently, the dynamical model has been generalized to improve global stability. In this paper we review the original stochastic model and both nonlinear models of Random Early Detection with a two-fold objective: (i) illustrate how a random model can be smoothed out to a deterministic one through data aggregation, and (ii) how this translation can shed light into complex processes such as the Internet data traffic. Furthermore, this paper contains new materials concerning the occurrence of chaos, bifurcation diagrams, Lyapunov exponents and global stability robustness with respect to control parameters. The results reviewed and reported here are expected to help design an active queue management algorithm in real conditions, that is, when system parameters such as the number of users and the round-trip time of the data packets change over time. The topic also illustrates the much-needed synergy of a theoretical approach, practical intuition and numerical simulations in engineering.
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