Desired features

[pnkraemer]

All of the following should be possible with pnfindiff at some point of the future, ideally, documented in example notebooks.

Basic scalar derivatives:

Evaluate the numerical derivative f'(x_0) of a function f: R -> R at point x_0

• [x] Central, forward, backward schemes
• [x] Variable derivative order
• [x] Variable method order
• [x] Numerical noise
• [x] Custom grid
• [x] Batching over x_0
• [x] Batching over x_0 with a non-uniform grid
• [x] Batching over f
• [x] Polynomial vs exponentiated quadratic kernel

Basic multivariate derivatives

Evaluate the numerical derivative (D f) (x_0) of a function f: R^n -> R^m at point x_0

• [x] Partial derivatives via schemes
• [x] Gradients via a tensor-product approach and using all of the above
• [ ] Jacobians and Hessians via a tensor-product approach and using all of the above
• [ ] Gradients, via an n-dimensional prior
• [ ] (Mixed) partial derivatives for n-dimensional priors
• [ ] Jacobians (forward and reverse mode?), Hessians, (JVPs?) via a n-dimensional prior

Advanced features

• [x] Uncertainty calibration
• [ ] Control over stencils -> which point is on boundary, etc. (https://github.com/maroba/findiff)
• [ ] Computation of (L f)(Xn) with FD. This is not the same as batched evaluation, because for instance, boundary nodes may use forward schemes, and central nodes use central schemes. (https://numpy.org/doc/stable/reference/generated/numpy.gradient.html)
• [ ] Support for symbolic computation
• [ ] Support for sparse matrices
• [ ] A bunch of kernels: matern, (inverse) multiquadrics, thin-plate splines
• [ ] Boundary conditions?
• [ ] Advanced differential operators: Divergence, Laplacian, curl
• [ ] Polar coordinates
• [ ] complex differentials? integer-valued derivatives? (e.g. grad(f, allow_int=True, holomorphic=True))
• [ ] Unsymmetric and symmetric collocation (with examples and docs)