Physics-Informed Neural Network with Fourier Features for Radiation Transport in Heterogeneous Media

Figures & data

Fig. 1. PiNN example: The neural network consists of two input neurons (in the case of a 2-D phase-space), followed by $n$ hidden layers, each with $m$ neurons, and an output layer for the scalar variable, solution of the PDE $\mathcal{F}(\psi (x,\mu ))=q$. The residual of the PDE at sampled points in the phase-space volume is built, as well as the boundary residual. Both residuals are then in the definition of a loss function whose minimization process trains the parameters of the neural network.

Fig. 2. Radiation transport PiNN in direction ${\mu }_d$ with Fourier Features added.

Fig. 3. Small example of a neural network.

TABLE I Problem 1 Definition

$x$	${\mathrm{\Sigma }}_a$	${\mathrm{\Sigma }}_s$	$q$
[0,5]	5	0	5

Fig. 4. Scalar flux for Problem 1: PiNN and FD solutions.

TABLE II Problem 2 Definition

$x$	${\mathrm{\Sigma }}_a$	${\mathrm{\Sigma }}_s$	$q$
[0,0.5]	2.5	2.5	5
[0.5,1]	2.5	2.5	0

Fig. 5. Scalar flux for Problem 2: PiNN and FD solutions.

Fig. 6. Training error with scattering.

TABLE III Problem 3 Definition

$x$	${\mathrm{\Sigma }}_a$	${\mathrm{\Sigma }}_s$	$q$
[0,1]	50	0	5
[1,5]	1	0	0

Fig. 7. Problem 3 with Fourier Features: PiNN and FD solutions.

Fig. 8. Problem 3 without Fourier Features: PiNN and FD solutions.

Fig. 9. Single PiNN loss with and without Fourier Features.

Fig. 10. Internal sampled point approaches.

TABLE IV Problem 4 Definition

$x$	${\mathrm{\Sigma }}_a$	${\mathrm{\Sigma }}_s$	$q$
[0,1]	50	0	5
[1,5]	0	0	0

Fig. 11. Scalar flux Problem 4: PiNN and FD solutions.

TABLE V Problem 4’ Definition

$x$	${\mathrm{\Sigma }}_a$	${\mathrm{\Sigma }}_s$	$q$
[0,1]	50	0	5
[1,2]	0	0	0

Fig. 12. Angular flux Problem 4’: PiNN and FD solutions.

TABLE VI Reed’s Problem Definition (Problem 5)

$x$	${\mathrm{\Sigma }}_a$	${\mathrm{\Sigma }}_s$	$q$
[0,2]	50	0	50
[2,3]	5	0	0
[3,5]	0	0	0
[5,6]	0.1	0.9	1
[6,8]	0.1	0.9	0

Fig. 13. Scalar flux Problem 5: PiNN and FD solutions.

Fig. 14. Source iteration convergence error Problem 5.