Simplifying Physics Informed Neural Networks in case of periodicity to address low quality and sparse data while solving differential equations : an application in fluid dynamics.

To address most scientific problems, two approaches usually stand : mathematical modelling or experimentation. Physics Informed Neural Network [1] (PINN) is a recent numerical method that closes this gap using multi-layer perceptrons that approximate physical quantities. This allows resolution of ill-posed problems with a light formalism by optimizing residuals of differential equations and fitting provided data. However classical PINN can have difficulties to converge. To tackle that issue in the presence of periodic dynamics, we introduce ModalPINN where a truncated Fourier decomposition is enforced directly into the neural network’s structure. Application of this technique is given for a periodic flow using simulated experimental data. Gains up to 2 orders of magnitude in precision and robustness regarding added noise or delay are observed.

[1] M. Raissi, P. Perdikaris, and G. E. Karniadakis, J. Comput. Phys. 378, 686 (2019).