Model reduction and linearisation of a physiological dynamical system using deep learning-based Koopman analysis.

Computational models give us insights into physiological processes and help to develop novel treatment strategies in the biomedical field. However, their complexity and non-linearity make them computationally expensive and challenging to analyse. Reduced or linearised representations of these models can help overcome these issues. A potential application of such models addresses the design of physiological control algorithms for vagus nerve stimulation (VNS). Here models will be essential to control stimulation and regulate the cardiovascular system function improving clinical outcomes. In this thesis steps to address this problem have been investigated using a deep neural network approach to Koopman analysis (DNN-KA). DNN-KA uses deep learning to globally linearise dynamics on a low-dimensional manifold by identifying non-linear coordinates with a modified autoencoder. The training optimised a combined cost function comprising a reconstruction-, prediction-, and linearity term. The general procedure involves data gathering, normalization, splitting, network training, and quantification of prediction accuracy. The practical work consisted of three main parts. First, to ensure validity of the implementation, the results reported in literature for a simple non-linear toy model were reproduced. Second, the approach was applied to the Fitz-Hugh-Nagumo (FHN) model. Random search was performed on hyperparameters to optimise prediction accuracy which included, but were not limited to, autoencoder width (w), number of layers in the autoencoder (d). Forty-eight DNN were trained on a dataset generated by numerically solving the FHN model for 1000 initial conditions and 30 timesteps. Finally, a preliminary exploration was conducted on applying the DNN-KA to identify the linear embeddings of selected hemodynamic and electrophysiological signals using a static model architecture. Therefore, a dataset was generated by numerically solving a 0-D cardiovascular model for 5000 initial conditions, extracting a segment of 71 timesteps centred at the onset of VNS. For the toy model, the training- and validation errors were in the magnitude of 10-7 which is in accordance with the results reported in literature, and the relative root mean square error (RRMSE) was 0.22 %. For the FHN model, the training- and validation errors and RRMSE, for the best DNN architecture (w=128, d=3), were 8.8⋅10-5, 1.1⋅10-4, and 14.1 %, respectively. For the cardiovascular model, the training and validation errors and the RRMSE for the best combination of input parameters were 1.9⋅10-5, 5⋅10-5, and 7.2 %. For the FHN model, the prediction accuracy and linearity were satisfying. However, the linearisation of the model came at a cost of computational time subjected to about a 5-fold increase, suggesting that model order reduction and linearisation may have competing effects on computational cost. Furthermore, it cannot be excluded that there might be better-performing DNN architectures, as the number of architectures tested was limited. The results for the cardiovascular system model were also promising, however, further work is needed, which includes an hyperparameter optimisation and expansion of the number of used hemodynamic signals. Overall, the results are encouraging and add to the evidence supporting the usefulness of DNN-KA and its application to identify linear embeddings for hemodynamic signals is novel. This approach can be of particular interest for the development of physiological control algorithms in the future.