Machine learning from limited data - Predicting biological dynamics under a time-varying external input

Aug 5, 2026

Reservoir computing is used to predict highly stochastic biological dynamics from limited observations, focusing on cell-shape dynamics driven by a time-varying external input. An echo state network learns the steady-state behavior from very small data and captures transient timescales using only four observations. With this “dynamic twin,” the method extends beyond direct forecasting to infer statistics of cell-shape dynamics under unobserved conditions and to quantify determinism versus stochasticity.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages6

Reading Time15 min

AccessFree to download

Introduction

Reservoir computing for time-varying biological data

Dynamics twin concept

Model and Data Background

Echo state network reservoir dynamics

Hyperparameters, training and listening phases

Ridge regression and prediction stage

External parameter tagging

Machine learning from limited data: Predicting biological dynamics under a time-varying external input Hoony Kang, Keshav Srinivasan, and Wolfgang Losert University of Maryland, College Park, Maryland 20742 (Dated: September 17, 2024) Reservoir computing (RC) is known as a powerful machine learning approach for learning complex dynamics from limited data. Here, we use RC to predict highly stochastic dynamics of cell shapes. We find that RC is able to predict the steady state climate from very limited data. Furthermore, the RC learns the timescale of transients from only four observations. We find that these capabilities of the RC to act as a dynamic twin allows us to also infer important statistics of cell shape dynamics of unobserved conditions. arXiv:2408.07998v2 [[physics.bio-ph](physics.bio-ph)] 15 Sep 2024 I. INTRODUCTION Machine learning has been applied extensively in analyzing and predicting complex dynamics. It is particularly important for biological systems since they cannot be fully captured with simple mathematical models, and since biological systems are generally non-stationary, i.e. continuously evolve their behavior in time. One Machine Learning architecture known to learn rapidly enough to deal with such time varying data is Reservoir Computing (RC) [1–4] . In this paper, we implement a RC, specifically an echo state network [5], to analyze and predict the boundary motion of a large migratory cell that is subject to a time varying electric field E. We provide this context to the RC by equipping it with a dynamic input channel that supposes a parameter of the dynamical system[6, 7] . This allows us to assess the reservoir’s predictability for each parameter value, and its adaptability as the parameter is switched mid-prediction. We also implement a parallel architecture that incorporates knowledge of the input network structure to improve performance[8, 9] . We use the RC with its additional parameter channel as a ‘dynamics twin’ (i.e. as a simulated system that can be used to explore changes in parameters or previously unobserved conditions of the dynamic system) for the E-guided cell boundary motion. The dynamic twin allows us to (1) estimate the degree to which the measured boundary motion of a cell is deterministic versus stochastic, (2) assess transition characteristics for sudden changes in parameters, and (3) obtain the statistics of interpolated system states. II. MODEL AND DATA BACKGROUND The equation describing the node states r(t) in a reservoir is r (t + ∆t) =αr(t) + (1 − α)tanh 􀀂Ar (t) + Win u (t)+ b1􀀃 (1) where the activation function tanh[·] is taken to apply to each component of the vector equation in its argument separately. The strengths of the nonzero elements are set by ‘hyperparameters’ that scale the matrices’ elements. Elements of Win ∈ RNr ×Nu are chosen from a uniform distribution, i.e. Winij ∈ [−σ,σ] where σ ∈ R+ is the input scaling. The sparse adjacency matrix A ∈ RNr ×Nr is scaled by its spectral radius ρ to ensure the echo state property. Nr is the number of nodes in the reservoir and Nu is the input dimension of u. b is some constant bias, and 1 denotes a vector of 1s. The state of the reservoir is then cast back onto the original spatial domain of the input data via a linear mapping ˜u : RNr → RNu defined by ˜u(t) = Wout r (t) (2) where Wout ∈ RNu ×Nx is the output matrix that maps the reservoir states back onto the input domain. As ˜u(t+∆t) assumes the prediction of udata after one time-step ∆t, the goal is to obtain the matrix Wout such that ˜u(t) best approximates u (t) for chosen hyperparameters. To do this, the reservoir is initiated into a ‘listening phase’ during the first few time-steps of feeding in the input, i.e. for τlisten steps of ∆t. Here, u (t) = utrain (t), thereby entraining the states r (t) to the dynamics of the input data. As this stage is used solely to synchronize the nodes with the data, these reservoir states a

FAQs

How does reservoir computing handle time-varying biological data in this work?

The approach uses an echo state network that is designed to learn dynamics from inputs that evolve over time, including a dynamic input channel to track parameter changes during prediction.

What is the role of the dynamic twin in predicting cell shape dynamics?

The dynamic twin is a simulated system derived from the reservoir, enabling inference of unobserved-condition statistics, assessment of determinism versus stochasticity, and estimation of transition characteristics after parameter changes.

How are the transient timescales and steady-state behavior learned from limited data?

The method shows that the steady state climate can be predicted from very limited data, while transient timescales are learned using only four observations, leveraging the reservoir’s learned dynamics representation.

Aug 5, 2026

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