In my free time, I like to play first person shooter games, GUI development and CAD modelling. I am an advocate of open-source software, at the same time I am an active member of r/LinuxSucks. I also like to travel to remote places.
In recent years, physics-informed neural networks (PINN) have been used to solve stiff-PDEs mostly in the 1D and 2D spatial domain. PINNs still experience issues solving 3D problems, especially, problems with conflicting boundary conditions at adjacent edges and corners. These problems have discontinuous solutions at edges and corners that are difficult to learn for neural networks with a continuous activation function. In this review paper, we have investigated various PINN frameworks that are designed to solve stiff-PDEs. We took two heat conduction problems (2D and 3D) with a discontinuous solution at corners as test cases. We investigated these problems with a number of PINN frameworks, discussed and analysed the results against the FEM solution. It appears that PINNs provide a more general platform for parameterisation compared to conventional solvers. Thus, we have investigated the 2D heat conduction problem with parametric conductivity and geometry separately. We also discuss the challenges associated with PINNs and identify areas for further investigation.
@article{sharma2023stiff,title={Stiff-PDEs and Physics-Informed Neural Networks},author={Sharma, Prakhar and Evans, Llion and Tindall, Michelle and Nithiarasu, Perumal},journal={Archives of Computational Methods in Engineering},publisher={Springer},pages={1--30},year={2023},doi={10.1007/s11831-023-09890-4},dimensions={true}}
Hyperparameter selection for physics-informed neural networks (PINNs)–Application to discontinuous heat conduction problems
Prakhar Sharma, Llion Evans, Michelle Tindall, and 1 more author
Numerical Heat Transfer, Part B: Fundamentals, 2023
In recent years, physics-informed neural networks (PINNs) have emerged as an alternative to conventional numerical techniques to solve forward and inverse problems involving partial differential equations (PDEs). Despite its success in problems with smooth solutions, implementing PINNs for problems with discontinuous boundary conditions (BCs) or discontinuous PDE coefficients is a challenge. The accuracy of the predicted solution is contingent upon the selection of appropriate hyperparameters. In this work, we performed hyperparameter optimization of PINNs to find the optimal neural network architecture, number of hidden layers, learning rate, and activation function for heat conduction problems with a discontinuous solution. Our aim was to obtain all the settings that achieve a relative L2 error of 10% or less across all the test cases. Results from five different heat conduction problems show that the optimized hyperparameters produce a mean relative L2 error of 5.60%.
@article{sharma2023hyperparameter,title={Hyperparameter selection for physics-informed neural networks (PINNs)--Application to discontinuous heat conduction problems},author={Sharma, Prakhar and Evans, Llion and Tindall, Michelle and Nithiarasu, Perumal},year={2023},journal={Numerical Heat Transfer, Part B: Fundamentals},publisher={Taylor \& Francis},pages={1--15},doi={10.1080/10407790.2023.2264489},dimensions={false}}
