Application and Comparison of Parameterized Neural Ordinary Differential Equations for Single Parameter Engineering Models

The study tested the parameterized neural ordinary differential equation (PNODE) framework with a physical system exhibiting only advective phenomenon. Existing deep learning methods have difficulty learning multiple dynamic, continuous time processes. PNODE encodes the input data and initial parameter into a set of reduced states within the latent space. Then the reduced states are fitted to a system of ordinary differential equations. The outputs from the model are then decoded back to the data space for a desired input parameter and time. The application of the PNODE formalism to different types of physical systems is important to test the methods robustness. The linear advection data was generated through a high-fidelity numerical tool for multiple velocity parameters. The PNODE code was modified for the advection dataset, whose temporal domain and spatial discretization varied from the original study configuration. The L2 norm between the reconstruction and surrogate model and the reconstruction plots were used to analyze the PNODE model performance. The model reconstructions presented mixed results. For a temporal domain of 20-time units, where multiple advection cycles were completed for each advection speed, the reconstructions did not agree with the surrogate model. For a reduced temporal domain of 5-time units, the reconstructions and surrogate models were in close agreement. Near the end of the temporal domain, deviations occurred likely resulting from the accumulation of numerical errors. Note, over the 5-time units, smaller advection speed parameters were unable to complete a cycle. The behavior for the 20-time units highlighted potential issues with imbalanced datasets and repeated features. The 5-time unit model illustrates PNODEs adaptability to this class of problems when the dataset is better posed.