PyTorch

PyTorch is a software framework for numerical tensor operations and machine learning, including automatic differentiation and tools for training neural networks. Its array operations provide the calculations; its differentiation machinery records how a result depends on selected inputs. A computed gradient is distinct from the later optimization step that uses it.^[ref]

For example, this small Python calculation differentiates $y=x^2+3x$ at $x=2$:

import torch
x = torch.tensor(2.0, requires_grad=True)
y = x * x + 3 * x
y.backward()
print(x.grad)  # tensor(7.)

The result agrees with $dy/dx=2x+3$. The call to backward() computes and accumulates derivatives; it does not itself change x to minimize y. In a training loop, parameter updates and clearing old gradients are separate operations.^[ref]

A useful consequence of its dynamically constructed computation is that executed control flow determines the recorded graph. Recording a new graph on the next execution makes it possible to differentiate the computation that actually ran. Disabling gradient tracking or detaching a value changes that connection and must be deliberate.^[ref]