Parameter Estimation of Mechanistic Differential Equations via Neural Differential Equations
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Persisting trends of increased data availability and refined user-friendly tools to model large datasets has encouraged renewed interest in constructing data-driven models to solve real-world problems, with much success. In particular, Neural Ordinary Differential Equations (Neural ODEs) have recently demonstrated the ability to interpolate dynamic data of arbitrary nonlinearity. However, data-driven models are often cursed with limited interpretability and fail to obey physical laws governing many engineering and scientific applications. Alternatively, mechanistic models, which use prior knowledge based on physical laws or domain expertise, often require less data and observe better extrapolation performance, significantly reducing the experimental overhead to track system relationships. However, building mechanistic models may become computationally intractable when the model’s differential equations are strongly nonlinear or a good initial guess for the parameter values is unavailable. This work demonstrates how Neural ODEs can be used as a data-driven means to a mechanistic end—namely, estimating the parameter values in mechanistic differential equations. Using a 2-stage, or indirect, approach the Neural ODE can be trained to accurately estimate the derivative profiles of the system states. Then in the second step, the derivative and state estimates of the Neural ODE can be used to estimate the parameters of the original mechanistic model. In this presentation, the performance of the Neural ODE approach is characterized for scenarios of varying measurement noise and mechanistic model nonlinearity. Moreover, we compare the proposed Neural ODE approach with traditional direct approaches to regress differential equation models for examples ranging from ecology to chemical engineering.