Electromagnetic Simulation Tools + Automatic Differentiation. Code for the arxiv preprint Forward-Mode Differentiation of Maxwell's Equations.

(logo by @nagilmer)

ceviche provides two core electromagnetic simulation tools for solving Maxwell's equations:

• finite-difference frequency-domain (FDFD)

• finite-difference time-domain (FDTD)

Both are written in numpy / scipy and are compatible with the HIPS autograd package, supporting forward-mode and reverse-mode automatic differentiation.

This allows you to write code to solve your E&M problem, and then use automatic differentiation on your results.

As a result, you can do gradient-based optimization, sensitivity analysis, or plug your E&M solver into a machine learning model without the tedius process of deriving your derivatives analytically.

Let's saw we inject light at position source and measure its intensity at probe.

Between these two points, there's a box at location pos_box with permittivity eps.

We're interested in computing how the intensity measured changes with respect to eps.

With ceviche, we first write a simple function computing the measured intensity as a function of eps using FDFD

import autograd.numpy as np           # import the autograd wrapper for numpy
from ceviche import fdfd_ez as fdfd   # import the FDFD solver

# make an FDFD simulation
f = fdfd(omega, dl, eps_box, npml=[10, 10])

def intensity(eps):
    """ computes electric intensity at `probe` for a given box permittivity of `eps`

        source |-----| probe
            .  | eps |  .
               |_____|
    """

    # set the permittivity in the box region to the input argument
    fdfd.eps_r[box_pos] = eps

    # solve the fields
    Ex, Ey, Hz = f.solve(source)

    # compute the intensity at `probe`
    I = np.square(np.abs(Ex)) + np.square(np.abs(Ey))
    return = np.sum(I * probe)

Then, finding the derivative of the intensity is as easy as calling one function and evaluating at the current permittivity

# use autograd to differentiate `intensity` function
grad_fn = jacobian(intensity)

# then, evaluate it at the current value of `eps`
dI_deps = grad_fn(eps_curr)

Note that we didnt have to derive anything by hand for this specific situation!

Armed with this capability, we can now do things like gradient based optimization to maximize the intensity.

for _ in range(10):
    eps_current += step_size * dI_deps_fn(eps_current)

This becomes more powerful when you have several degrees of freedom, like in a topology optimization problem, or when your machine learning model involves running an FDFD or FDTD simulation.

ceviche is designed with simplicity and flexibility in mind and is meant to serve as a base package for building your projects from. Because of this -- with some exceptions -- it does not have simple interfaces for optimization, source or device creation, or visualization. While those things may be added later, for now you will need to build them yourself. Thankfully, because ceviche takes care of the hard parts, this can be relatively easy!

For some inspiration, see the examples directory.

For more user friendly features, check out our angler package. We plan to merge the two packages at a later date to give these automatic differentiation capabilities to angler.

There are many ways to install ceviche.

The easiest is by

pip install ceviche

But to install from a local copy, one can instead do

git clone https://github.com/twhughes/ceviche.git
pip install -e ceviche
pip install -r ceviche/requirements.txt

from the main directory.

Alternatively, just download it:

git clone https://github.com/twhughes/ceviche.git

and then import the package from within your python script

import sys
sys.path.append('path/to/ceviche')

The ceviche directory contains everything needed.

To get the FDFD and FDTD simulators, import directly from ceviche import fdtd, fdfd_ez, fdfd_hz, fdfd_ez_nl

To get the differentiation, import from ceviche import jacobian.

constants.py contains some constants EPSILON_0, C_0, ETA_0, Q_E, which are needed throughout the package

utils.py contains a few useful functions for plotting, autogradding, and various other things.

There are many demos in the examples directory, which will give you a good sense of how to use the package.

Tests are located in tests. To run, cd into tests and

python -m unittest

to run all or

python specific_test.py

to run a specific one. Some of these tests involve visual inspection of the field plots rather than error checking on values.

To run all of the gradient checking functions, run

bash tests/test_all_gradients.sh

If you use this for your research or work, please cite

@misc{1908.10507,
Author = {Tyler W Hughes and Ian A D Williamson and Momchil Minkov and Shanhui Fan},
Title = {Forward-Mode Differentiation of Maxwell's Equations},
Year = {2019},
Eprint = {arXiv:1908.10507},
}