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Modeling physiochemical relationships using dynamic data is a common task in fields throughout science and engineering. A common step in developing generalizable, mechanistic models is to fit unmeasured parameters to measured data. However, fitting differential equation-based models can be computation-intensive and uncertain due to the presence of nonlinearity, noise, and sparsity in the data, which in turn causes convergence to local minima and divergence issues. This work proposes a merger of machine learning (ML) and mechanistic approaches by employing ML models as a means to fit nonlinear mechanistic ordinary differential equations (ODEs). Using a two-stage indirect approach, neural ODEs are used to estimate state derivatives, which are then used to estimate the parameters of a more interpretable mechanistic ODE model. In addition to its computational efficiency, the proposed method demonstrates the ability of neural ODEs to better estimate derivative information than interpolating methods based on algebraic data-driven models. Most notably, the proposed method is shown to yield accurate predictions even when little information is known about the parameters of the ODEs. The proposed parameter estimation approach is believed to be most advantageous when the ODE to be fit is strongly nonlinear with respect to its unknown parameters.
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This content will become publicly available on April 1, 2026
A Hybrid ODE-NN Framework for Modeling Incomplete Physiological Systems
This paper proposes a method to learn ap- proximations of missing Ordinary Differential Equations (ODEs) and states in physiological models where knowl- edge of the system’s relevant states and dynamics is in- complete. The proposed method augments known ODEs with neural networks (NN), then trains the hybrid ODE-NN model on a subset of available physiological measurements (i.e., states) to learn the NN parameters that approximate the unknown ODEs. Thus, this method can model an ap- proximation of the original partially specified system sub- ject to the constraints of known biophysics. This method also addresses the challenge of jointly estimating physio- logical states, NN parameters, and unknown initial condi- tions during training using recursive Bayesian estimation. We validate this method using two simulated physiolog- ical systems, where subsets of the ODEs are assumed to be unknown during the training and test processes. The proposed method almost perfectly tracks the ground truth in the case of a single missing ODE and state and performs well in other cases where more ODEs and states are missing. This performance is robust to input signal per- turbations and noisy measurements. A critical advantage of the proposed hybrid methodology over purely data-driven methods is the incorporation of the ODE structure in the model, which allows one to infer unobserved physiological states. The ability to flexibly approximate missing or inac- curate components in ODE models improves a significant modeling bottleneck without sacrificing interpretability.
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- PAR ID:
- 10639183
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- IEEE Transactions on Biomedical Engineering
- Volume:
- 72
- Issue:
- 4
- ISSN:
- 0018-9294
- Page Range / eLocation ID:
- 1377 to 1386
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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