Session: 08-04-04 Advanced Analytics

Paper Number: 81237

81237 - Physics-Informed Neural Network With Numerical Differentiation for Modelling Complex Fluid Dynamic Problems 

Physics-informed neural networks (PINNs) have been received increasing attention recently. The central idea of a PINN is to incorporate governing physical laws, typically differential equations, into a loss function which will be minimized during a training process. PINNs have been demonstrated for various physics, including heat transfer, fluid dynamics and electromagnetics. The method is particularly suitable for inverse problems as it does not require well-posed conditions strictly satisfied. Nevertheless, current developments of PINNs are still at an early stage. There remain major challenges preventing it to be trully practical.

One of the major challenges is the evaluation of differential operators in the governing equations, such as the Gradient and Laplacian in the Navier-Stokes equations. The current PINNs use automatic differentiation (AD) for this purpose (called a-PINN). The AD is a main engine behind the training of a neural network as it computes the gradient of the loss function. However, when the network is over-parameterized, which is often, insufficient collocation points make a-PINN susceptible to inaccurate solutions. The AD indeed returns the exact localized gradient of the solution; hence it is very sensitive to the change of the solution at every colocation point during the training. Often, one can see gradients varying over several orders of magnitude at the same point or among adjacent colocation points. Due to this local effect, a-PINN may fulfill the underlying differential equations well at all collocation points, leading to a near zero training loss, even when its solution is far different from the true solution.

To overcome this challenge, we employ the numerical differentiation (ND) techniques to evaluate the differential operators. This allows PINN (we call it n-PINN) to modulate gradient behavior at piece-wise local regions rather than isolated collocation points. As a result, n-PINNs learn the pattern in larger solution spaces even with sparsely sampled collocation points, although its accuracy depends on discretization and numerical scheme used. Numerical experiments on several fluid dynamic problems have demonstrated that n-PINN could provide more accurate and reliable solutions that a-PINN could not. n-PINN is also more efficient than a-PINN as it requires much less operation especially in big networks. Using the ND formulation, n-PINN will also benefits from the pool of numerical methods which is already well-established.

Presenting Author: My Ha Dao Institute of High Performance Computing

Authors:

My Ha Dao Institute of High Performance Computing
Pao-Hsiung Chiu Institute of High Performance Computing
Jian Cheng Wong Institute of High Performance Computing
Chin Chun Ooi Institute of High Performance Computing