Part of the XAD ecosystem — the fastest automatic differentiation library for C++. If this is useful to you, please star the core XAD repository.
XAD is a library designed for automatic differentiation, aimed at both beginners and advanced users. It is intended for use in production environments, emphasizing performance and ease of use. The library facilitates the computation of derivatives within computer programs, making the process efficient and straightforward for a wide range of mathematical functions, from simple arithmetic to complex calculations, ensuring accurate and automatic derivative computations.
The Python bindings for XAD offer the following features:
- Support for both forward and adjoint modes at the first order.
- Strong exception-safety guarantees.
- High performance, as demonstrated in extensive production use.
For more details and to integrate XAD into your projects, consult the comprehensive documentation.
Application Areas
Automatic differentiation has many application areas, for example:
- Machine Learning and Deep Learning: Training neural networks or other machine learning models.
- Optimization: Solving optimization problems in engineering and finance.
- Numerical Analysis: Enhancing numerical solution methods for differential equations.
- Scientific Computing: Simulating physical systems and processes.
- Risk Management and Quantitative Finance: Assessing and hedging risk in financial models.
- Computer Graphics: Optimizing rendering algorithms.
- Robotics: Improving control and simulation of robotic systems.
- Meteorology: Enhancing weather prediction models.
- Biotechnology: Modeling biological processes and systems.
Try it in your browser
Run these notebooks on Google Colab, with nothing to install:
| Notebook | |
|---|---|
| Quickstart: gradients in adjoint and forward mode, checked against finite differences |  |
| Option Greeks: Black-Scholes and Monte Carlo price with all Greeks in one adjoint pass | |
| Adjoint vs finite differences: gradient of a diffusion model as the number of parameters grows | |
Getting Started
Install:
pip install xad
Supports Python 3.10 to 3.14. The package version follows the XAD C++ core it is built on.
Calculate first-order derivatives in adjoint mode:
import xad.adj_1st as xadj
# set independent variables
x0_ad = xadj.Real(1.0)
x1_ad = xadj.Real(1.5)
x2_ad = xadj.Real(1.3)
x3_ad = xadj.Real(1.2)
with xadj.Tape() as tape:
# and register them
tape.registerInput(x0_ad)
tape.registerInput(x1_ad)
tape.registerInput(x2_ad)
tape.registerInput(x3_ad)
# start recording derivatives
tape.newRecording()
# calculate the output
y = x0_ad + x1_ad - x2_ad * x3_ad
# register and seed adjoint of output
tape.registerOutput(y)
y.derivative = 1.0
# compute all other adjoints
tape.computeAdjoints()
# output results
print(f"y = {y}")
print(f"first order derivatives:\n")
print(f"dy/dx0 = {x0_ad.derivative}")
print(f"dy/dx1 = {x1_ad.derivative}")
print(f"dy/dx2 = {x2_ad.derivative}")
print(f"dy/dx3 = {x3_ad.derivative}")
Complex numbers work the same way: Complex has active real and imaginary parts, and
xad.cmath mirrors Python's cmath (see samples/complex.py):
import xad.adj_1st as xadj
import xad.cmath
x, y = xadj.Real(0.5), xadj.Real(1.2)
with xadj.Tape() as tape:
tape.registerInput(x)
tape.registerInput(y)
tape.newRecording()
z = xadj.Complex(x, y)
w = z * xad.cmath.exp(z)
tape.registerOutput(w.real)
w.real.derivative = 1.0
tape.computeAdjoints()
print(f"dRe(w)/dx = {x.derivative}, dRe(w)/dy = {y.derivative}")
Second-order derivatives use nested modes, as in XAD's C++: xad.fwd_adj_2nd (forward over
adjoint) and xad.fwd_fwd_2nd. xad.value(x) and xad.derivative(x) return the inner first-order
parts, so xad.derivative(xad.derivative(x)) is a second derivative (see
samples/fwd_adj_2nd.py). For whole matrices, xad.jacobian and
xad.hessian take a function of a list of Reals:
import xad
from xad.math import sin, cos
def f(x):
return sin(x[0] * x[1]) - cos(x[1] * x[2]) - sin(x[2] * x[3]) - cos(x[3] * x[0])
print(xad.hessian(f, [1.0, 1.5, 1.3, 1.2]))
Checkpointing (in adj_1st and fwd_adj_2nd) replaces part of the recording with a
CheckpointCallback, whose computeAdjoint(tape) is called during computeAdjoints, for example
to recompute that part in a nested recording and save tape memory
(see samples/checkpointing.py). The tape keeps inserted callbacks
alive, and ScopedNestedRecording is a with block:
class Cb(xadj.CheckpointCallback):
def computeAdjoint(self, tape):
ybar = tape.getAndResetOutputAdjoint(self.y_slot)
x = xadj.Real(self.x_value)
tape.registerInput(x)
with xadj.ScopedNestedRecording(tape) as nested:
y = f(x) # recompute actively
tape.registerOutput(y)
y.derivative = ybar
nested.computeAdjoints()
nested.incrementAdjoint(self.x_slot, x.derivative)
# forward run: compute y passively, store x_value and the slots, then tape.insertCallback(cb)
External functions use the same interface: a function computed outside XAD (passively, or in
another library) enters the recording through a CheckpointCallback whose computeAdjoint
applies its hand-written adjoint (see samples/external_function.py):
class ExternalSum(xadj.CheckpointCallback):
def compute_external(self, xs):
tape = xadj.Tape.getActive()
self.x_slots = [x.getSlot() for x in xs]
y = xadj.Real(sum(x.value for x in xs)) # computed passively
tape.registerOutput(y)
self.y_slot = y.getSlot()
tape.insertCallback(self)
return y
def computeAdjoint(self, tape):
ybar = tape.getAndResetOutputAdjoint(self.y_slot)
for s in self.x_slots:
tape.incrementAdjoint(s, ybar) # d(sum)/dx_i = 1
NumPy Arrays
xad.adj_1st.Array and xad.fwd_1st.Array hold n-dimensional arrays of active values in C++,
so whole-array operations run as one C++ loop each instead of one Python call per element
(see samples/numpy_arrays.py):
import numpy as np
import xad.adj_1st as xadj
with xadj.Tape() as tape:
x = xadj.Array([0.1, 0.2, 0.3])
tape.registerInputs(x)
tape.newRecording()
y = np.sum(np.arange(1.0, 4.0) * np.exp(x))
tape.registerOutput(y)
y.derivative = 1.0
tape.computeAdjoints()
print(x.derivatives) # dy/dx as a float array
Supported are elementwise operators and ufuncs (arithmetic, comparisons, exp, log, sqrt,
sin, ...) with broadcasting, sum/prod/mean with axis, @/dot, where,
concatenate, and indexing, slicing and reshape as views. a[i, j] is the element Real
itself, np.asarray(a) gives the values, and np.asarray(a, dtype=object) the elements.
Unsupported NumPy functions raise TypeError rather than dropping the derivatives. Arrays need
NumPy >= 1.25 at runtime; it is not a build or import dependency of the package.
For more information, see the Documentation.
Building Extensions Against XAD
The package ships XAD as a shared library, with its C++ headers and a CMake package config.
C++ extension modules linking it share the active tape with xad.adj_1st.Tape
(per thread):
find_package(XAD CONFIG REQUIRED)
target_link_libraries(my_ext PRIVATE XAD::xad)
cmake -B build -DXAD_DIR=$(python -c "import xad; print(xad.get_cmake_dir())")
xad.get_include() and xad.get_library_dir() give the header and library directories.
Import xad before the extension so the library is loaded.
Related Projects
1 maintainer
Metadata
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Release files for xad 2.2.0
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| Download URL | xad-2.2.0-cp310-cp310-macosx_11_0_arm64.whl |
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Transparency logRelease files / xad-2.2.0-cp310-cp310-macosx_10_9_x86_64.whl
| Download URL | xad-2.2.0-cp310-cp310-macosx_10_9_x86_64.whl |
|---|---|
| Size | 1.4 MB |
| Tags | CPython 3.10 macOS 10.9+ x86-64 |
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3512c7158360518d86e8d43721a6066baffaaa24068d46280bbac025700523d8
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Transparency logSecurity for xad 2.2.0
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