Convolutional Autoencoder and Koopman Operator-Based Markov Chaining for Time Series Forecasting of a Large-Scale Dissipative Particle Dynamics System for Thrombosis Modelling

<b>Background:</b> Cardiovascular disease (CVD) is the world's leading cause of death, responsible for over 17.9 million deaths annually. The majority of CVD-related deaths are caused by thrombosis, the formation of a blood clot in a vein or artery that impedes the circulatory system. Therefore, it is necessary to create accurate platelet and flow dynamics models to simulate the thrombosis process effectively. Multiscale modeling (MSM) combines various spatiotemporal resolutions in order to achieve high accuracy while simultaneously reducing the computational cost. Preceding works defined MSM models involving Dissipative Particle Dynamics (DPD) for the Poiseuille flow of blood in the vessel and Coarse-Grained Molecular Dynamics (CGMD) for the fluid-platelet interface. However, these simulations are highly computationally intensive due to the large amount of DPD calculations performed on fluid particles at high resolutions with negligible benefits. It is necessary to find an accelerated solution to the fluid modeling aspect of the simulation.<br/><br/><b>Methods: </b>We define the DPD system as a 3D velocity field on a defined particle plane 391.2 μm long x 21.3 μm wide. The velocity field is represented by a tensor of size (4200, 244, 3) for a total of 3074400 elements. The simulation is discretized in timestep increments of 250 μs with a corresponding state tensor at each timestep. In order to utilize a Koopman Operator to model time series evolution, we derived a series of observable functions that accurately reflect the state of the system in a lower dimensional space than the state tensor. We derived these functions by implementing a Convolutional Autoencoder (CAE) to learn a 256-element latent representation of the state tensor and consider spatial relationships in the velocity field; this latent vector acts as the observable function. We used Extended Dynamic Mode Decomposition (EDMD) to derive an approximated Koopman Operator (aKO). EDMD is performed by taking the product of a matrix of latent vectors for each state tensor time step joined as columns with the Moore-Penrose pseudoinverse of a time-shifted variant of this matrix. In order to model the time series evolution, the observable function at a discrete time is multiplied by the Koopman operator to predict the next observable function. This process forms a Markov chain where the model's current state is the only consideration in predicting the future state.<br/><br/><b>Results:</b> CAE-aKO resulted in a speed increase of 390x for single-step prediction, a value that increased linearly to 3570x for nine steps. CAE-aKO also achieved high accuracy with an average RMSE value of 0.1372 across 3 million elements and an average MAE of 0.1065. These averages were calculated across predictions ranging from 1-9 timesteps (250 μs - 2.25 ms) into the future. We attribute the majority of the prediction error to flaws in the CAE’s learned latent vectors, as the aKO presented low errors in latent vector time series prediction when the latent vectors are not converted back to state tensors.<br/><br/><b>Discussion and Future Work:</b> We assessed the viability of a non-physics-guided approach to time series forecasting of a large-scale system with complex spatiotemporal relationships. CAE-aKO learned an accurate underlying representation of the state tensor and was able to accurately model the evolution of the system with minimal drops in accuracy as multiple steps of prediction were performed. Future work will focus on refining the CAE architecture to minimize reconstruction loss and then use CAE-aKO to produce new data upon which to calculate aKOs for larger time steps, allowing for greater speedups in modeling methodology that allow for simulations to move from predicting on time scales of milliseconds to larger scales of seconds and minutes.<br/><br/><b>Acknowledgments:</b> This project was supported by Stony Brook University’s Garcia Center for Polymer Research and the Garcia Scholars Program.