# Neural Networks with Cheap Differential Operators

arXiv: https://arxiv.org/abs/1912.03579 · HTML: https://mlsanity.com/paper/1912.03579/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Dec 2019.

## Built on by (most cited first, 18 of 18)

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## Abstract (arXiv)

Gradients of neural networks can be computed efficiently for any architecture, but some applications require differential operators with higher time complexity. We describe a family of restricted neural network architectures that allow efficient computation of a family of differential operators involving dimension-wise derivatives, used in cases such as computing the divergence. Our proposed architecture has a Jacobian matrix composed of diagonal and hollow (non-diagonal) components. We can then modify the backward computation graph to extract dimension-wise derivatives efficiently with automatic differentiation. We demonstrate these cheap differential operators for solving root-finding subproblems in implicit ODE solvers, exact density evaluation for continuous normalizing flows, and evaluating the Fokker--Planck equation for training stochastic differential equation models.