Extremization to Fine Tune Physics Informed Neural Networks for Solving Boundary Value Problems
This work addresses computational bottlenecks in solving stiff and high-dimensional PDEs for scientific computing applications, representing an incremental advancement in PINN methods.
The paper tackles the challenge of fast and accurate training of physics-informed neural networks (PINNs) for boundary value problems by combining deep neural networks with Extreme Learning Machines, achieving significant improvements in training and inference times, such as with a proposed Reduced TFC method that generalizes to complex geometries.
We propose a novel method for fast and accurate training of physics-informed neural networks (PINNs) to find solutions to boundary value problems (BVPs) and initial boundary value problems (IBVPs). By combining the methods of training deep neural networks (DNNs) and Extreme Learning Machines (ELMs), we develop a model which has the expressivity of DNNs with the fine-tuning ability of ELMs. We showcase the superiority of our proposed method by solving several BVPs and IBVPs which include linear and non-linear ordinary differential equations (ODEs), partial differential equations (PDEs) and coupled PDEs. The examples we consider include a stiff coupled ODE system where traditional numerical methods fail, a 3+1D non-linear PDE, Kovasznay flow and Taylor-Green vortex solutions to incompressible Navier-Stokes equations and pure advection solution of 1+1 D compressible Euler equation. The Theory of Functional Connections (TFC) is used to exactly impose initial and boundary conditions (IBCs) of (I)BVPs on PINNs. We propose a modification to the TFC framework named Reduced TFC and show a significant improvement in the training and inference time of PINNs compared to IBCs imposed using TFC. Furthermore, Reduced TFC is shown to be able to generalize to more complex boundary geometries which is not possible with TFC. We also introduce a method of applying boundary conditions at infinity for BVPs and numerically solve the pure advection in 1+1 D Euler equations using these boundary conditions.