Optimizing differential equations to fit data and predict outcomes

Frank, Steven A.

doi:10.1002/ece3.9895

Cited by 4 publications

(1 citation statement)

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“…NODEs have already been used as a powerful tool in several applications such as forecasting power demand [30], and modelling the dynamics of lithium batteries [31] and single-cell transcriptomic dynamics [32]. In ecology, NODEs have already been used to infer community interactions from time-series data of two [33,34] and three [35] species but have not been applied to study the dynamics of multispecies communities.…”

Section: Introductionmentioning

confidence: 99%

Using neural ordinary differential equations to predict complex ecological dynamics from population density data

Arroyo-Esquivel,

Klausmeier,

Litchman

2024

J. R. Soc. Interface.

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Simple models have been used to describe ecological processes for over a century. However, the complexity of ecological systems makes simple models subject to modelling bias due to simplifying assumptions or unaccounted factors, limiting their predictive power. Neural ordinary differential equations (NODEs) have surged as a machine-learning algorithm that preserves the dynamic nature of the data (Chen et al. 2018 Adv. Neural Inf. Process. Syst. ). Although preserving the dynamics in the data is an advantage, the question of how NODEs perform as a forecasting tool of ecological communities is unanswered. Here, we explore this question using simulated time series of competing species in a time-varying environment. We find that NODEs provide more precise forecasts than autoregressive integrated moving average (ARIMA) models. We also find that untuned NODEs have a similar forecasting accuracy to untuned long-short term memory neural networks and both are outperformed in accuracy and precision by empirical dynamical modelling . However, we also find NODEs generally outperform all other methods when evaluating with the interval score, which evaluates precision and accuracy in terms of prediction intervals rather than pointwise accuracy. We also discuss ways to improve the forecasting performance of NODEs. The power of a forecasting tool such as NODEs is that it can provide insights into population dynamics and should thus broaden the approaches to studying time series of ecological communities.

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Section: Introductionmentioning

confidence: 99%

Using neural ordinary differential equations to predict complex ecological dynamics from population density data

Arroyo-Esquivel,

Klausmeier,

Litchman

2024

J. R. Soc. Interface.

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Optimizing differential equations to fit data and predict outcomes

Frank

2023

Ecology and Evolution

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Many scientific problems focus on observed patterns of change or on how to design a system to achieve particular dynamics. Those problems often require fitting differential equation models to target trajectories. Fitting such models can be difficult because each evaluation of the fit must calculate the distance between the model and target patterns at numerous points along a trajectory. The gradient of the fit with respect to the model parameters can be challenging to compute. Recent technical advances in automatic differentiation through numerical differential equation solvers potentially change the fitting process into a relatively easy problem, opening up new possibilities to study dynamics. However, application of the new tools to real data may fail to achieve a good fit. This article illustrates how to overcome a variety of common challenges, using the classic ecological data for oscillations in hare and lynx populations. Models include simple ordinary differential equations (ODEs) and neural ordinary differential equations (NODEs), which use artificial neural networks to estimate the derivatives of differential equation systems. Comparing the fits obtained with ODEs versus NODEs, representing small and large parameter spaces, and changing the number of variable dimensions provide insight into the geometry of the observed and model trajectories. To analyze the quality of the models for predicting future observations, a Bayesian-inspired preconditioned stochastic gradient Langevin dynamics (pSGLD) calculation of the posterior distribution of predicted model trajectories clarifies the tendency for various models to underfit or overfit the data. Coupling fitted differential equation systems with pSGLD sampling provides a powerful way to study the properties of optimization surfaces, raising an analogy with mutationselection dynamics on fitness landscapes.

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