Physics-Informed Neural Networks and Functional Interpolation for Initial Value Problems with Applications to Integro-Differential and Stiff Differential Equations

Abstract

This dissertation aims to show the development and adaptation of the Extreme Theory of Functional Connections (X-TFC) framework, a Physics-Informed Neural Network (PINN) combined with Theory of Functional Connections (TFC), to solve Initial Value Problems (IVPs) modeled by differential equations (DEs). In particular, this research focuses on two branches of IVPs: Linear Integro-Differential Equations and Stiff Systems of Non-Linear Differential Equations. The first types of IVPs are DEs where an integral of the unknown function is present, i.e., Integro-Differential Equations. To compute the integral of the unknown function two approaches are used. The first one is the analytical integration of the unknown function's approximation, which is represented by the TFC's constrained expression (CE). The second one approximates the definite integral of the unknown function via Gauss--Legendre quadrature. This approach is used to solve problems arising from the Boltzmann Partial Integro-Differential Equation for Transport, such as Radiative Transfer and Rarefied-Gas Dynamics problems. The second types of IVPs are large-scale Stiff Systems of non-linear ODEs, mathematical models governing a wide range of real-world problems such as chemistry, biology, epidemiology, engineering, neuroscience, financial systems, and so on. For this reason, recently, there has been a resurgence in the interest in developing new numerical methods capable of solving large time-scale stiff problems. Regular PINNs frameworks are proven to be not accurate (or even not capable) to solve problems when the dynamics are particularly complex (e.g., stiff). This is due to unbalanced back-propagated gradients during the model training. This dissertation shows how X-TFC with a domain decomposition technique overcomes the difficulties of regular PINNs in solving systems of stiff IVPs, and it outperforms the state-of-the-art numerical methods in terms of accuracy and computational times.