Data to Differential Equations - Discovering Mathematical Models in Continuum Systems

Abstract

Mathematical modeling of complex systems encompassing multi-scale, multi-physics phenomena is an arduous task. Discovery and development of these models have changed very little over the past hundreds of years. This process involves tedious mathematical formulations, requiring complex derivations followed by validation against observed experimental data. The abundance of data available in the modern era combined with increased computational power facilitates a new approach for model development using data-driven techniques. Instead of validating a model with the observable data, the data itself may be used to identify the governing equations for the phenomena being studied.

Furthermore, the existing governing equations are typically solved using numerical methods that exact a high computational cost when high-fidelity results are required. To combat this, models of lower fidelity with significant approximations allow for faster computation at the expense of lower accuracy. New-wave, data-driven equation discovery can improve modeling of complex phenomena by discovering reduced order models that may not be easily derived with traditional techniques. Similarly, data-driven approaches may be used to develop surrogate models like neural networks, which are not interpretable, but can accurately approximate a function with fewer computational resources. Ultimately, these data-driven reduced order and surrogate models may simultaneously maintain accuracy while reducing computational cost.

In this thesis, we explore data-driven methods for the discovery of differential equations to model physical phenomena using a range of approaches including manual derivations, system identification, and neural networks. We developed a reduced order model of blood flow through stenosed coronary arteries, which may be used to aid physicians in making real-time decisions to improve patient outcomes. We considered standard approaches for model reduction followed by use of those equations to inform a system identification approach. These identified models are fast (running in seconds on a personal computer), interpretable, and provide improvements compared to existing low order models. Next, we explored phase field models of fracture for which we developed a variational system identification approach to identify the phenomena governing the fracture model. This method reduces the computational cost of fine-tuning a phase field model for fracture. Robustness in the face of noisy data and temporal subsampling was explored. Finally, a new data-driven approach was developed using non-local calculus on finite-weighted graphs to create a new neural network architecture that included differential operators within the layers, providing the capability to represent a differential equation. Direct representation enables one to use a surrogate model with additional mathematical interpretability and therefore accuracy. This method also enables the use of a smaller neural network, reducing the computational cost of training. In each of these applications, we utilized data to identify governing equations or surrogate models for complex, continuum systems.