# Disentangling Computation in Multi-Task Neural Networks with the Green's Operator

arXiv: https://arxiv.org/abs/2609.40292 · HTML: https://mlsanity.com/paper/2609.40292/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 6 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 3 of 3)

- [Lyapunov spectra of chaotic recurrent neural networks](https://mlsanity.com/paper/2006.02427/) (2020)
- [Finite-time Lyapunov exponents of deep neural networks](https://mlsanity.com/paper/2306.12548/) (2023)
- [The Global Empirical NTK: Self-Referential Bias and Dimensionality of Gradient Descent Learning](https://mlsanity.com/paper/2605.08746/) (2026)

## Abstract (arXiv)

How is computation organized and reused across tasks and time in a trained recurrent network? Most analyses emphasize the geometry of neural activity, dynamical motifs, or local perturbation growth. We instead study the network's global first-order perturbation response. The finite-horizon Green's operator maps perturbations at each source along a trajectory to their downstream state-space responses and therefore directly represents perturbation routing. Simple reductions of this operator provide task-to-task and time-to-time views of the same computation, while matrix-free products make these views accessible without constructing the full operator. In a flexible multitask recurrent network, task reductions reveal structured reuse of known computational motifs, while temporal reductions reveal causal pathways and how they emerge during training. Our main point is simple: the Green's operator provides a global response geometry for mapping the organization of learned dynamical computation.