SymPy#
Some useful SymPy pages:
Defining the amplitude model in terms of SymPy#
Parameters would become a mapping of Symbols to initial, suggested values and dynamics would be a mapping of Symbols to ‘suggested’ expressions. Intensity will be the eventual combined expression.
There needs to be one symbol \(x\) that represents the four-momentum input:
As an example, let’s create an AmplitudeModel with an intensity that is a sum of Gaussians. Each Gaussian here takes the rôle of a dynamics function:
model = AmplitudeModel()
N_COMPONENTS = 3
for i in range(1, N_COMPONENTS + 1):
mu = sp.Symbol(Rf"\mu_{i}")
sigma = sp.Symbol(Rf"\sigma_{i}")
model.initial_values.update({
mu: float(i),
sigma: 1 / (2 * i),
})
gauss = sp.exp(-((x - mu) ** 2) / (sigma**2)) / (sigma * sp.sqrt(2 * sp.pi))
dyn_symbol = sp.Symbol(Rf"\mathrm{{dyn}}_{i}")
model.dynamics[dyn_symbol] = gauss
coherent_sum = sum(model.dynamics)
model.intensity = coherent_sum
model.initial_values
{\mu_1: 1.0,
\sigma_1: 0.5,
\mu_2: 2.0,
\sigma_2: 0.25,
\mu_3: 3.0,
\sigma_3: 0.16666666666666666}
model.intensity
\[\displaystyle \mathrm{dyn}_1 + \mathrm{dyn}_2 + \mathrm{dyn}_3\]
Dynamics are inserted into the intensity expression of the model:
model.intensity.subs(model.dynamics)
\[\displaystyle \frac{\sqrt{2} e^{- \frac{\left(- \mu_{3} + x\right)^{2}}{\sigma_{3}^{2}}}}{2 \sqrt{\pi} \sigma_{3}} + \frac{\sqrt{2} e^{- \frac{\left(- \mu_{2} + x\right)^{2}}{\sigma_{2}^{2}}}}{2 \sqrt{\pi} \sigma_{2}} + \frac{\sqrt{2} e^{- \frac{\left(- \mu_{1} + x\right)^{2}}{\sigma_{1}^{2}}}}{2 \sqrt{\pi} \sigma_{1}}\]
And, for evaluating, the ‘suggested’ initial parameter values are inserted:
model.intensity.subs(model.dynamics).subs(model.initial_values)
\[\displaystyle \frac{1.0 \sqrt{2} e^{- 4.0 \left(x - 1.0\right)^{2}}}{\sqrt{\pi}} + \frac{2.0 \sqrt{2} e^{- 64.0 \left(0.5 x - 1\right)^{2}}}{\sqrt{\pi}} + \frac{3.0 \sqrt{2} e^{- 324.0 \left(0.333333333333333 x - 1\right)^{2}}}{\sqrt{\pi}}\]
Here’s a small helper function to plot this model:
Show code cell source
def plot_model(model: AmplitudeModel) -> None:
total_plot = sp.plotting.plot(
model.intensity.subs(model.dynamics).subs(model.initial_values),
(x, 0, 4),
show=False,
line_color="black",
)
p1 = sp.plotting.plot(
model.dynamics[sp.Symbol(R"\mathrm{dyn}_1")].subs(model.initial_values),
(x, 0, 4),
line_color="red",
show=False,
)
p2 = sp.plotting.plot(
model.dynamics[sp.Symbol(R"\mathrm{dyn}_2")].subs(model.initial_values),
(x, 0, 4),
line_color="blue",
show=False,
)
p3 = sp.plotting.plot(
model.dynamics[sp.Symbol(R"\mathrm{dyn}_3")].subs(model.initial_values),
(x, 0, 4),
line_color="green",
show=False,
)
total_plot.extend(p1)
total_plot.extend(p2)
total_plot.extend(p3)
total_plot.show()
plot_model(model)
Now we can couple parameters like this:
model.initial_values[sp.Symbol(R"\sigma_3")] = 1
plot_model(model)
And it’s also possible to insert custom dynamics:
Implementation in TensorWaves#
Credits @spflueger
1. Create a double gaussian amp with SymPy#
When building the model, we should be careful to pass the parameters as arguments as well, otherwise frameworks like jax can’t determine the gradient.
import math
import sympy as sp
x, A1, mu1, sigma1, A2, mu2, sigma2 = sp.symbols("x, A1, mu1, sigma1, A2, mu2, sigma2")
gaussian1 = (
A1 / (sigma1 * sp.sqrt(2.0 * math.pi)) * sp.exp(-((x - mu1) ** 2) / (2 * sigma1))
)
gaussian2 = (
A2 / (sigma2 * sp.sqrt(2.0 * math.pi)) * sp.exp(-((x - mu2) ** 2) / (2 * sigma2))
)
gauss_sum = gaussian1 + gaussian2
gauss_sum
\[\displaystyle \frac{0.398942280401433 A_{1} e^{- \frac{\left(- \mu_{1} + x\right)^{2}}{2 \sigma_{1}}}}{\sigma_{1}} + \frac{0.398942280401433 A_{2} e^{- \frac{\left(- \mu_{2} + x\right)^{2}}{2 \sigma_{2}}}}{\sigma_{2}}\]
2. Convert this expression to a function using lambdify#
TensorFlow as backend:
import inspect
tf_gauss_sum = sp.lambdify(
(x, A1, mu1, sigma1, A2, mu2, sigma2), gauss_sum, "tensorflow"
)
print(inspect.getsource(tf_gauss_sum))
def _lambdifygenerated(x, A1, mu1, sigma1, A2, mu2, sigma2):
return (0.398942280401433*A1*exp(-1/2*pow(-mu1 + x, 2)/sigma1)/sigma1 + 0.398942280401433*A2*exp(-1/2*pow(-mu2 + x, 2)/sigma2)/sigma2)
NumPy as backend:
numpy_gauss_sum = sp.lambdify((x, A1, mu1, sigma1, A2, mu2, sigma2), gauss_sum, "numpy")
print(inspect.getsource(numpy_gauss_sum))
def _lambdifygenerated(x, A1, mu1, sigma1, A2, mu2, sigma2):
return (0.398942280401433*A1*exp(-1/2*(-mu1 + x)**2/sigma1)/sigma1 + 0.398942280401433*A2*exp(-1/2*(-mu2 + x)**2/sigma2)/sigma2)
Jax as backend:
from jax import numpy as jnp
from jax import scipy as jsp
jax_gauss_sum = sp.lambdify(
(x, A1, mu1, sigma1, A2, mu2, sigma2),
gauss_sum,
modules=(jnp, jsp.special),
)
print(inspect.getsource(jax_gauss_sum))
def _lambdifygenerated(x, A1, mu1, sigma1, A2, mu2, sigma2):
return (0.398942280401433*A1*exp(-1/2*(-mu1 + x)**2/sigma1)/sigma1 + 0.398942280401433*A2*exp(-1/2*(-mu2 + x)**2/sigma2)/sigma2)
3. Natively create the respective packages#
import math
import tensorflow as tf
def gaussian(x, A, mu, sigma):
return (
A
/ (sigma * tf.sqrt(tf.constant(2.0, dtype=tf.float64) * math.pi))
* tf.exp(
-tf.pow(
-tf.constant(0.5, dtype=tf.float64) * (x - mu) / sigma,
2,
)
)
)
def native_tf_gauss_sum(x_, A1_, mu1_, sigma1_, A2_, mu2_, sigma2_):
return gaussian(x_, A1_, mu1_, sigma1_) + gaussian(x_, A2_, mu2_, sigma2_)
# @jx.pmap
def jax_gaussian(x, A, mu, sigma):
return (
A
/ (sigma * jnp.sqrt(2.0 * math.pi))
* jnp.exp(-((-0.5 * (x - mu) / sigma) ** 2))
)
def native_jax_gauss_sum(x_, A1_, mu1_, sigma1_, A2_, mu2_, sigma2_):
return jax_gaussian(x_, A1_, mu1_, sigma1_) + jax_gaussian(x_, A2_, mu2_, sigma2_)
4. Compare performance#
import numpy as np
parameter_values = (1.0, 0.0, 0.1, 2.0, 2.0, 0.2)
rng = np.random.default_rng(0)
np_x = rng.uniform(-1, 3, 10000)
tf_x = tf.constant(np_x)
def evaluate_with_parameters(function):
def wrapper():
return function(np_x, *(parameter_values))
return wrapper
def call_native_tf():
func = native_tf_gauss_sum
params = tuple(tf.Variable(v, dtype=tf.float64) for v in parameter_values)
def wrapper():
return func(tf_x, *params)
return wrapper
import timeit
from jax.config import config
config.update("jax_enable_x64", True)
print(
"sympy tf lambdify",
timeit.timeit(evaluate_with_parameters(tf_gauss_sum), number=100),
)
print(
"sympy numpy lambdify",
timeit.timeit(evaluate_with_parameters(numpy_gauss_sum), number=100),
)
print(
"sympy jax lambdify",
timeit.timeit(evaluate_with_parameters(jax_gauss_sum), number=100),
)
print("native tf", timeit.timeit(call_native_tf(), number=100))
print(
"native jax",
timeit.timeit(evaluate_with_parameters(native_jax_gauss_sum), number=100),
)
sympy tf lambdify 0.22086703500099247
sympy numpy lambdify 0.02661015900048369
sympy jax lambdify 0.24337401299999328
native tf 0.25517284799934714
native jax 0.2992750530011108
5. Handling parameters#
Some options:
5.1 Changing parameter values#
Can be done in the model itself…
But how can the values be propagated to the AmplitudeModel?
Well, if an amplitude model only defines parameters with a name and the values are supplied in the function evaluation, then everything is decoupled and there are no problems.
5.2 Changing parameter names#
Names can be changed in the sympy AmplitudeModel. Since this sympy model serves as the source of truth for the Function, all things generated from this model will reflect the name changes as well.
But going even further, since the Parameters are passed into the functions as arguments, the whole naming becomes irrelevant anyways.
tf_var_A1 = tf.Variable(1.0, dtype=tf.float64) <- does not carry a name!!
5.3 Coupling parameters#
This means that one parameter is just assigned to another one?
result = evaluate_with_parameters(jax_gauss_sum)()
result
DeviceArray([0.60618145, 2.03309932, 3.59630909, ..., 0.26144946,
3.05430146, 2.88912312], dtype=float64)
np_x
array([ 2.86815263, 2.51926301, 1.7962957 , ..., 0.81910382,
1.67313811, -0.25404634])
import matplotlib.pyplot as plt
plt.hist(np_x, bins=100, weights=result);
parameter_values = (1.0, 0.0, 0.1, 2.0, 2.0, 0.1)
result = evaluate_with_parameters(jax_gauss_sum)()
plt.hist(np_x, bins=100, weights=result);
6. Exchange a gaussian with some other function#
This should be easy if you know the exact expression that you want to replace:
\[\displaystyle \sin{\left(a x \right)} + \cos{\left(b x \right)}\]
expr.subs(sp.sin(a * x), C)
\[\displaystyle C + \cos{\left(b x \right)}\]
7. Matrix operations?#
from sympy.physics.quantum.dagger import Dagger
spin_density = sp.MatrixSymbol("rho", 3, 3)
amplitudes = sp.Matrix([[1 + sp.I], [2 + sp.I], [3 + sp.I]])
dummy_intensity = sp.re(
Dagger(amplitudes) * spin_density * amplitudes,
evaluate=False,
# evaluate=False is important otherwise it generates some function that cant
# be lambdified anymore
)
dummy_intensity
\[\begin{split}\displaystyle \operatorname{re}{\left(\left[\begin{matrix}1 - i & 2 - i & 3 - i\end{matrix}\right] \rho \left[\begin{matrix}1 + i\\2 + i\\3 + i\end{matrix}\right]\right)}\end{split}\]
tf_intensity = sp.lambdify(
spin_density,
dummy_intensity,
modules=(tf,),
)
print(inspect.getsource(tf_intensity))
def _lambdifygenerated(rho):
return (real(matmul(matmul(constant([[1 - 1j, 2 - 1j, 3 - 1j]]), rho), constant([[1 + 1j], [2 + 1j], [3 + 1j]]))))
real0 = tf.constant(0, dtype=tf.float64)
real1 = tf.constant(1, dtype=tf.float64)
intensity_result = tf_intensity(
np.array([
[
tf.complex(real1, real0),
tf.complex(real0, real0),
-tf.complex(real0, real1),
],
[
tf.complex(real0, real0),
tf.complex(real1, real0),
tf.complex(real0, real0),
],
[
tf.complex(real0, real1),
tf.complex(real0, real0),
tf.complex(real1, real0),
],
]),
)
intensity_result
<tf.Tensor: shape=(1, 1), dtype=float64, numpy=array([[13.]])>