PINN (Physics Informed Neural Network) is a deep learning based technique for solving PDEs (partial differential equations), ODEs (oridnary differential equations) which are an integral part of engineering and computational sciences. It is guided by physical laws and data. It aims to find parameters of neural network in order to approximate solution function of differential equations. To achieve this, prior knowledge of system equations (PDEs, ODEs) and data are made part of loss function which is minimised to find neural network with suitable parameters. For solving inverse problems (system identification) the unknown parameters of the system are made part of neural network and is obtained like any other parameters of neural network after training. This project is divided in a two parts. In first study, Lame parameters are identified using tanh activation function. After that, six activation functions are analysed on the basis of minimum loss, training time and convergence order for different error norms. In this study softplus activation function performs best while elu (another activation function) performs worst.
Physics Informed Neural Network (PINN) is a different class of Artificial Neural Network (ANN) which aims to include governing equations into the neural network model. The loss function in PINN gives representation to known data, boundary conditions as well as governing equations. It requires very less data for training as compared to traditional data-driven neural networks. It can be used to solve ODE’s, PDE’s when used as a forward solver and can also identify parameters of differential equations (system identification).
SciANN is a high level API (Application Programming Interface) written in python which is used for scientific computing including PINN. It runs Keras and Tensorflow in backend. Follow this link to learn more: https://www.sciann.com/
This concept is used for second part of the study. A priori estimates for a numerical method are predictions on how the solution –or the error of the numerical solution against the exact (very accurate reference solution)– converges. For sufficiently regular problems the following a priori estimates hold in the L2-norm, the H1-norm and the energy-norm A in the simulation domain B:
The elasticity problem selected for the parameter identification study by PINN is an unit square element having constraints as shown in figure below. The origin lies at the bottom left corner of the element. This element is subjected to body forces in x and y direction as given by following equations.
The loss function is calculated by following equation:
Lame parameter were identified with very high accuracy (0.002% for λ and 9.22E-04% for µ)
Following equations were used for L2 norm error calculation and energy norm error calculation:
After comparison of several activation functions on the basis of convergence order, better fit of power laws, minimum loss value and training time, Softplus performed best in all these aspects while ELU performed worst in training time and minimum loss value.
- Parameter_ident : contains code for 16X16 grid size and weight folder for trained weights. Grid size can be customized for amy other values.
- Activation_fun_comparison : Contain primary code with two other folders called gnuplot_data (code and data for plotting convergence gnuplots) and weights (contain trained weights). To learn more about gnuplots click here.
[1] SciANN Documentation., year = 2021, url = https://www.sciann.com, urldate = 2021- 12-14.
[2] E. Haghighat and R. Juanes. Sciann: A keras/tensorflow wrapper for scientific compu�tations and physics-informed deep learning using artificial neural networks. Computer Methods in Applied Mechanics and Engineering, 373:113552, 2021.
[3] E. Haghighat, M. Raissi, A. Moure, H. Gomez, and R. Juanes. A deep learning framework for solution and discovery in solid mechanics, 2020.