Device-to-device Differentiable FDTD for Solving Electromagnetic Inverse Scattering
Overview
This tutorial introduces the method for solving electromagnetic inverse problems provided by MindSpore Elec based on device-to-device differentiable FDTD. The process of solving Maxwell’s equations by the finite-difference time-domain (FDTD) method is equivalent to a recurrent convolutional network (RCNN). The device-to-device differentiable FDTD can be obtained by rewriting the update process with the differentiable operator of MindSpore. Compared with the data-driven black-box model, the solution process of the differentiable FDTD method strictly satisfies the constraints of Maxwell’s equations. Using MindSpore gradient-ased optimizer, differentiable FDTD can solve various EM inverse problems.
This example is for GPU processors and you can download the full sample code here: https://gitee.com/mindspore/mindscience/tree/r0.2.0/MindElec/examples/AD_FDTD/fdtd_forward
Maxwell’s Equations
Active Maxwell’s equations are the classical control equations for electromagnetic simulation, which are a set of partial differential equations describing the relationship between electric and magnetic fields and charge density and current density, in the following form:
where \(\epsilon,\mu\) are the absolute permittivity and absolute magnetic permeability respectively. \(J(x, t)\) is the excitation source in the electromagnetic simulation process, which is usually expressed in the form of a port pulse. This is approximated in a mathematical sense as the point source expressed in Dirac functional form, which can be expressed as:
where \(x_0\) is the excitation source position and \(g(t)\) is the functional expression of the pulse signal.
Problem Description
This case solves the electromagnetic inverse scattering in two-dimensional TM mode. The two plastids are located inside the rectangular area. Four excitation sources (red triangles) and eight observation points (green dots) are set on the outside of the solution area, as shown in the following figure:
The process that MindSpore Elec solves the problem is as follows:
Obtain the time domain electric field values at the observation points with traditional numerical methods to create training datasets and evaluation data.
Define the excitation source location, the observation point location and the solution area.
Define the time domain waveform of the excitation source.
Construct differentiable FDTD networks.
Train the network and evaluate the solution results.
Import Dependencies
Import the modules and interfaces that this tutorial depends on:
import os
import argparse
import numpy as np
from mindspore import nn
import matplotlib.pyplot as plt
from src import transverse_magnetic, EMInverseSolver
from src import zeros, tensor, vstack, elu
from src import Gaussian, CFSParameters, estimate_time_interval
from src import BaseTopologyDesigner
Loading the Dataset
Load the time-domain electric field values at each observation point calculated by conventional numerical methods (e.g., FDTD), as well as the true value of the relative dielectric constant.
def load_labels(nt, dataset_dir):
"""
Load labels of Ez fields and epsr.
Args:
nt (int): Number of time steps.
dataset_dir (str): Dataset directory.
Returns:
field_labels (Tensor, shape=(nt, ns, nr)): Ez at receivers.
epsr_labels (Tensor, shape=(nx, ny)): Ground truth for epsr.
"""
field_label_path = os.path.join(dataset_dir, 'ez_labels.npy')
field_labels = tensor(np.load(field_label_path))[:nt]
epsr_label_path = os.path.join(dataset_dir, 'epsr_labels.npy')
epsr_labels = tensor(np.load(epsr_label_path))
return field_labels, epsr_labels
Defining the Excitation Source Location, Observation Point Location and Solution Area
Inheriting the BaseTopologyDesiger class, users can quickly define electromagnetic inverse scattering problems. Users can set the solution area, excitation source location and observation point location in the member functions generate_object, update_sources and get_outputs_at_each_step, respectively. Taking this problem as an example, we define the electromagnetic inverse scattering problem as follows.
class InverseDomain(BaseTopologyDesigner):
"""
InverseDomain with customized mapping and source locations for user-defined problems.
"""
def generate_object(self, rho):
"""Generate material tensors.
Args:
rho (Parameter): Parameters to be optimized in the inversion domain.
Returns:
epsr (Tensor, shape=(self.cell_nunbers)): Relative permittivity in the whole domain.
sige (Tensor, shape=(self.cell_nunbers)): Conductivity in the whole domain.
"""
# generate background material tensors
epsr = self.background_epsr * self.grid
sige = self.background_sige * self.grid
# ---------------------------------------------
# Customized Differentiable Mapping
# ---------------------------------------------
epsr[30:70, 30:70] = self.background_epsr + elu(rho, alpha=1e-2)
return epsr, sige
def update_sources(self, *args):
"""
Set locations of sources.
Args:
*args: arguments
Returns:
jz (Tensor, shape=(ns, 1, nx+1, ny+1)): Jz tensor.
"""
sources, _, waveform, _ = args
jz = sources[0]
jz[0, :, 20, 50] = waveform
jz[1, :, 50, 20] = waveform
jz[2, :, 80, 50] = waveform
jz[3, :, 50, 80] = waveform
return jz
def get_outputs_at_each_step(self, *args):
"""Compute output each step.
Args:
*args: arguments
Returns:
rx (Tensor, shape=(ns, nr)): Ez fields at receivers.
"""
ez, _, _ = args[0]
rx = [
ez[:, 0, 25, 25],
ez[:, 0, 25, 50],
ez[:, 0, 25, 75],
ez[:, 0, 50, 25],
ez[:, 0, 50, 75],
ez[:, 0, 75, 25],
ez[:, 0, 75, 50],
ez[:, 0, 75, 75],
]
return vstack(rx)
It should be noted that when mapping the variable to be optimized rho, to the relative permittivity epsr in generate_object, choosing the right mapping relationship can greatly speed up the solver convergence. The mapping relationship used in this case is background_epsr + elu(rho, alpha=1e-2), which guarantees that no unphysical dielectric constant will appear during the solution.
Defining the Excitation Source Time Domain Waveform
The time domain waveform of the excitation source in this case is a Gaussian pulse. FDTD uses the leap-frog scheme to update the electric and magnetic fields separately, and the excitation source in this case is a voltage source, so the time domain waveform value of the excitation source on the half time step should be calculated.
def get_waveform_t(nt, dt, fmax):
"""
Compute waveforms at time t.
Args:
nt (int): Number of time steps.
dt (float): Time interval.
fmax (float): Maximum freuqency of Gaussian wave
Returns:
waveform_t (Tensor, shape=(nt, ns, nr)): Waveforms.
"""
t = (np.arange(0, nt) + 0.5) * dt
waveform = Gaussian(fmax)
waveform_t = waveform(t)
return waveform_t
Constructing the Differentiable FDTD Network
This case solves the electromagnetic inverse scattering problem in two-dimensional TM mode. The process of solving Maxwell’s equations by the finite-difference in time domain (FDTD) method is equivalent to a recurrent convolutional network (RCNN). When the CFS-PML is used to truncate the infinite region, the update process of the \(n\)th time step of the two-dimensional FDTD in TM mode is as follows:
The device-to-device differentiable FDTD is obtained by rewriting the update process of the FDTD with the differentiable operator of MindSpore. The computation process within each time step is as follows:
class FDTDLayer(nn.Cell):
"""
One-step 2D TM-Mode FDTD.
Args:
cell_lengths (tuple): Lengths of Yee cells.
cpmlx_e (Tensor): Updating coefficients for electric fields in the x-direction CPML.
cpmlx_m (Tensor): Updating coefficients for magnetic fields in the x-direction CPML.
cpmly_e (Tensor): Updating coefficients for electric fields in the y-direction CPML.
cpmly_m (Tensor): Updating coefficients for magnetic fields in the y-direction CPML.
"""
def __init__(self,
cell_lengths,
cpmlx_e, cpmlx_m,
cpmly_e, cpmly_m,
):
super(FDTDLayer, self).__init__()
dx = cell_lengths[0]
dy = cell_lengths[1]
self.cpmlx_e = cpmlx_e
self.cpmlx_m = cpmlx_m
self.cpmly_e = cpmly_e
self.cpmly_m = cpmly_m
# operators
self.dx_oper = ops.Conv2D(out_channel=1, kernel_size=(2, 1))
self.dy_oper = ops.Conv2D(out_channel=1, kernel_size=(1, 2))
self.dx_wghts = tensor([-1., 1.]).reshape((1, 1, 2, 1)) / dx
self.dy_wghts = tensor([-1., 1.]).reshape((1, 1, 1, 2)) / dy
self.pad_x = ops.Pad(paddings=((0, 0), (0, 0), (1, 1), (0, 0)))
self.pad_y = ops.Pad(paddings=((0, 0), (0, 0), (0, 0), (1, 1)))
def construct(self, jz_t, ez, hx, hy, pezx, pezy, phxy, phyx,
ceze, cezh, chxh, chxe, chyh, chye):
"""One-step forward propagation
Args:
jz_t (Tensor): Source at time t + 0.5 * dt.
ez, hx, hy (Tensor): Ez, Hx, Hy fields.
pezx, pezy, phxy, phyx (Tensor): CPML auxiliary fields.
ceze, cezh (Tensor): Updating coefficients for Ez fields.
chxh, chxe (Tensor): Updating coefficients for Hx fields.
chyh, chye (Tensor): Updating coefficients for Hy fields.
Returns:
hidden_states (tuple)
"""
# -------------------------------------------------
# Step 1: Update H's at n+1/2 step
# -------------------------------------------------
# compute curl E
dezdx = self.dx_oper(ez, self.dx_wghts) / self.cpmlx_m[2]
dezdy = self.dy_oper(ez, self.dy_wghts) / self.cpmly_m[2]
# update auxiliary fields
phyx = self.cpmlx_m[0] * phyx + self.cpmlx_m[1] * dezdx
phxy = self.cpmly_m[0] * phxy + self.cpmly_m[1] * dezdy
# update H
hx = chxh * hx - chxe * (dezdy + phxy)
hy = chyh * hy + chye * (dezdx + phyx)
# -------------------------------------------------
# Step 2: Update E's at n+1 step
# -------------------------------------------------
# compute curl H
dhydx = self.pad_x(self.dx_oper(hy, self.dx_wghts)) / self.cpmlx_e[2]
dhxdy = self.pad_y(self.dy_oper(hx, self.dy_wghts)) / self.cpmly_e[2]
# update auxiliary fields
pezx = self.cpmlx_e[0] * pezx + self.cpmlx_e[1] * dhydx
pezy = self.cpmly_e[0] * pezy + self.cpmly_e[1] * dhxdy
# update E
ez = ceze * ez + cezh * ((dhydx + pezx) - (dhxdy + pezy) - jz_t)
hidden_states = (ez, hx, hy, pezx, pezy, phxy, phyx)
return hidden_states
The device-to-device differentiable FDTD network is as follows:
class ADFDTD(nn.Cell):
"""2D TM-Mode Differentiable FDTD Network.
Args:
cell_numbers (tuple): Number of Yee cells in (x, y) directions.
cell_lengths (tuple): Lengths of Yee cells.
nt (int): Number of time steps.
dt (float): Time interval.
ns (int): Number of sources.
designer (BaseTopologyDesigner): Customized Topology designer.
cfs_pml (CFSParameters): CFS parameter class.
init_weights (Tensor): Initial weights.
Returns:
outputs (Tensor): Customized outputs.
"""
def __init__(self,
cell_numbers,
cell_lengths,
nt, dt,
ns,
designer,
cfs_pml,
init_weights,
):
super(ADFDTD, self).__init__()
self.nx = cell_numbers[0]
self.ny = cell_numbers[1]
self.dx = cell_lengths[0]
self.dy = cell_lengths[1]
self.nt = nt
self.ns = ns
self.dt = tensor(dt)
self.designer = designer
self.cfs_pml = cfs_pml
self.rho = ms.Parameter(
init_weights) if init_weights is not None else None
self.mur = tensor(1.)
self.sigm = tensor(0.)
if self.cfs_pml is not None:
# CFS-PML Coefficients
cpmlx_e, cpmlx_m = self.cfs_pml.get_update_coefficients(
self.nx, self.dx, self.dt, self.designer.background_epsr.asnumpy())
cpmly_e, cpmly_m = self.cfs_pml.get_update_coefficients(
self.ny, self.dy, self.dt, self.designer.background_epsr.asnumpy())
cpmlx_e = tensor(cpmlx_e.reshape((3, 1, 1, -1, 1)))
cpmlx_m = tensor(cpmlx_m.reshape((3, 1, 1, -1, 1)))
cpmly_e = tensor(cpmly_e.reshape((3, 1, 1, 1, -1)))
cpmly_m = tensor(cpmly_m.reshape((3, 1, 1, 1, -1)))
else:
# PEC boundary
cpmlx_e = cpmlx_m = tensor([0., 0., 1.]).reshape((3, 1))
cpmly_e = cpmly_m = tensor([0., 0., 1.]).reshape((3, 1))
# FDTD layer
self.fdtd_layer = FDTDLayer(
cell_lengths, cpmlx_e, cpmlx_m, cpmly_e, cpmly_m)
# auxiliary variables
self.dte = tensor(dt / epsilon0)
self.dtm = tensor(dt / mu0)
# material parameters smoother
self.smooth_kernel = 0.25 * ones((1, 1, 2, 2))
self.smooth_oper = ops.Conv2D(
out_channel=1, kernel_size=2, pad_mode='pad', pad=1)
def construct(self, waveform_t):
"""
ADFDTD-based forward propagation.
Args:
waveform_t (Tensor, shape=(nt,)): Time-domain waveforms.
Returns:
outputs (Tensor): Customized outputs.
"""
# ----------------------------------------
# Initialization
# ----------------------------------------
# constants
ns, nt, nx, ny = self.ns, self.nt, self.nx, self.ny
dt = self.dt
# material grid
epsr, sige = self.designer.generate_object(self.rho)
# delectric smoothing
epsrz = self.smooth_oper(epsr[None, None], self.smooth_kernel)
sigez = self.smooth_oper(sige[None, None], self.smooth_kernel)
# set materials on the interfaces
(epsrz, sigez) = self.designer.modify_object(epsrz, sigez)
# non-magnetic & magnetically lossless material
murx = mury = self.mur
sigmx = sigmy = self.sigm
# updating coefficients
ceze, cezh = fcmpt(self.dte, epsrz, sigez)
chxh, chxe = fcmpt(self.dtm, murx, sigmx)
chyh, chye = fcmpt(self.dtm, mury, sigmy)
# hidden states
ez = create_zero_tensor((ns, 1, nx + 1, ny + 1))
hx = create_zero_tensor((ns, 1, nx + 1, ny))
hy = create_zero_tensor((ns, 1, nx, ny + 1))
# CFS-PML auxiliary fields
pezx = zeros_like(ez)
pezy = zeros_like(ez)
phxy = zeros_like(hx)
phyx = zeros_like(hy)
# set source location
jz_t = zeros_like(ez)
# ----------------------------------------
# Update
# ----------------------------------------
outputs = []
t = 0
while t < nt:
jz_t = self.designer.update_sources(
(jz_t,), (ez,), waveform_t[t], dt)
# RNN-Style Update
(ez, hx, hy, pezx, pezy, phxy, phyx) = self.fdtd_layer(
jz_t, ez, hx, hy, pezx, pezy, phxy, phyx,
ceze, cezh, chxh, chxe, chyh, chye)
# Compute outputs
outputs.append(
self.designer.get_outputs_at_each_step((ez, hx, hy)))
t = t + 1
outputs = hstack(outputs)
return outputs
Model Training
Define the differentiable FDTD network, loss function, optimizer, iteration steps and learning rate, then define the electromagnetic inverse scattering solver object solver and call the solve interface for solving.
# define fdtd network
fdtd_net = transverse_magnetic.ADFDTD(
cell_numbers, cell_lengths, nt, dt, ns,
inverse_domain, cpml, rho_init)
# define sovler for inverse problem
epochs = options.epochs
lr = options.lr
loss_fn = nn.MSELoss(reduction='sum')
optimizer = nn.Adam(fdtd_net.trainable_params(), learning_rate=lr)
solver = EMInverseSolver(fdtd_net, loss_fn, optimizer)
# solve
solver.solve(epochs, waveform_t, field_labels)
Evaluating the Solution Results
After the solution is finished, the eval interface is called to evaluate the PSNR and SSIM of the inversion results.
epsr, _ = solver.eval(epsr_labels)
The relative permittivity of PSNR and SSIM obtained by inversion based on the above method are:
[epsr] PSNR: 27.835317 dB, SSIM: 0.963564
The relative permittivity distribution obtained by the inversion is shown in the figure below:
