SymNum

What is SymNum?

SymNum is a Python package that acts a bridge between NumPy and SymPy, providing a NumPy-like interface that can be used to symbolically define functions which take arrays as arguments and return arrays or scalars as values. A series of Autograd style functional differential operators are also provided to construct derivatives of symbolic functions, with the option to generate NumPy code to numerically evaluate these derivative functions.

Why use SymNum instead of Autograd or JAX?

SymNum is intended for use in generating the derivatives of 'simple' functions which compose a relatively small number of operations and act on small array inputs. By reducing interpreter overheads it can produce code which is cheaper to evaluate than corresponding Autograd or JAX functions (including those using JIT compilation) in such cases, and which can be serialised with the inbuilt Python pickle library allowing use for example in libraries which use multiprocessing to implement parallelisation across multiple processes.

The original motivating use case for SymNum was to allow automatically constructing the derivatives of the sorts of functions of low dimensional inputs which are commonly used as toy examples to demonstrate inference and optimisation algorithms. In these cases while manually deriving and implementing derivatives is generally possible, this can still be labourious and error prone, and distract from the purpose of giving a simple show case of an algorithm. On the other hand the derivative functions produced by Autograd and JAX in such cases are often much slower than manual implementations. SymNum tries to fill this gap by providing the flexibility and ease of use that comes from automatic differentiation while still being efficient for small toy examples.

Doesn't SymPy already have array support and allow export of NumPy functions?

Yes: SymNum is mainly a convenience wrapper around functionality already provided by SymPy to make it easier to use for those already familiar with NumPy and Autograd / JAX. Specifically SymPy has several inbuilt array like classes, which can be broadly split in to the array types defined in sympy.tensor.array and the matrix types defined in sympy.matrices.

Each of the inbuilt array and matrix classes supports some of the functionality of NumPy's core ndarray class, however both have some issues which means they don't provide an easy drop-in replacement, with for example matrix classes being limited to two-dimensions, while both the inbuilt array and matrix classes do not support the full broadcasting and operator overloading semantics of NumPy arrays. The SymbolicArray class in symnum.array aims to provide a more ndarray like interface, supporting broadcasting of elementwise binary arithmetic operations like *, /, + and -, elementwise NumPy ufunc-like mathematical functions like numpy.log via the symnum.numpy module, simple array contractions over potentially multiple axes with the sum and prod methods and matrix multiplication with the @ operator.

Similarly SymPy has extensive built in code generation features, including the lambdify function which supports generation of functions which operate on NumPy arrays. It can be non-trivial however to use these functions to generate code which perform indexing operations on array inputs, or to construct higher order functions which return closures. SymNum builds on top of the SymPy's code generation functionality to allow simpler generation of NumPy functions using such features.

Example

```Python import numpy as np import symnum.numpy as snp from symnum import namedarray, numpifyfunc, jacobian

Define a function using the symnum.numpy interface.

def func(x): return (snp.array([[1., -0.5], [-2., 3.]]) @ snp.array([snp.cos(-x[1]**2 + 3 * x[0]), snp.sin(x[0] - 1)]))

Create a named symbolic array to act as input and evaluate func symbolically.

x = named_array(name='x', shape=2) y = func(x)

Alternatively we can symbolically 'trace' func and use this to generate a

NumPy function which accepts ndarray arguments. To allow the tracing we

need to manually specify the shapes of the arguments to the function.

xnp = np.array([0.2, 1.1]) funcnp = numpifyfunc(func, x.shape) ynp = funcnp(xnp)

We can also use a similar approach to generate a NumPy function to evaluate

the Jacobian of func on ndarray arguments. The numpified function func_np

stores the symbolic function used to generate it and details of the argument

shapes and so we can pass it as a sole argument to jacobian without

specifying the argument shapes.

jacobfuncnp = jacobian(funcnp) dydxnp = jacobfuncnp(xnp) ```

See also the demo Jupyter notebook.

Current limitations

SymNum only supports a small subset of the NumPy API at the moment. A non-exhaustive list of things that don't currently work

• Indexed / sliced assignment to arrays e.g. a[i, j] = x and a[:, j] = y
• Matrix multiplication with @ of arrays with dimensions > 2.
• Linear algebra operations in numpy.linalg and FFT functions in numpy.fft.
• All scipy functions such as the special functions in scipy.special.
• Similar to the limitations on using Python control flow with the JIT transformation in JAX, the symbolic tracing of functions with SymNum requires that only control flows that does not depend on the value of array arguments is used.

Some of these are not fundamental limitations and SymNum's coverage will improve (pull requests are very welcome!), however as the focus is on allowing automatic generation of derivatives of simple functions of smallish arrays if your use case uses more complex NumPy features you are likely to find Autograd or JAX to be better bets.