Overview Tables

Tables on Automatic Differentiation

• Overview of Explicit Automatic Differentiation Rules for both scalar and tensor operations (forward- and reverse-mode)
• Overview of Implicit Automatic Differentiation Rules like root-finding, (non-)linear system solving, ODE and PDE integration or differentiable optimization (forward- and reverse-mode)

The tables below are experimental and not yet complete.

• Overview of 1D Convolution Automatic Differentiation Rules with different padding and stride options (only reverse-mode, the forward-mode is straightforward based on the matrix-vector multiplication rule)
• Overview of Convolution Automatic Differentiation Rules in various deep learning frameworks (only reverse-mode, the forward-mode is straightforward based on the matrix-vector multiplication rule): this differs from the above table in that:
  1. some frameworks (like PyTorch, TensorFlow or JAX) use cross-correlation instead of convolution
  2. real-world convolutions/cross-correlations often map multiple channels to multiple channels and do so for multiple samples (one batch) at once (that requires additional reordering of tensor axes for the reverse passes)
  3. the tensor layout convention (of batch, spatial and channel axes) can vary between frameworks

Tables on Physics-based deep learning

• Overview of learning setups for neural predictors like the classical t-step supervised, but also more that involve differentiable physics simulators
• Overview of corrector configurations for neural correctors that are trained to augment coarse solvers

Tables on Fast Fourier Transform (FFT)

• Relation of functions and their Fourier coefficients in 1d: shows, for instance
  1. how the coefficients are scaled
  2. that a real positive cosine has both positive real coefficients at positive and negative wavenumbers
  3. that a real positive sine has a negative imaginary coefficient at the positive wavenumber and a positive imaginary coefficient at the negative wavenumber
  4. how imaginary sines and cosines are represented
  5. how the Nyquist mode glitches at an even number of sampling points
  6. how higher modes are aliased
• Similar table for 2d: here, we additionally have:
  1. how the indexing scheme for np.meshgrid affects the order of the coefficients
  2. how the scaling of the coefficients changes (N scaling for zero mode, N/2 scaling for modes where one axis is the zero mode, N/4 scaling for all other modes (except the Nyquist mode at even numbers of sampling points))
  3. how the Nyquist mode glitches, especially if one axis has an even number of sampling points and the other an odd number