# SILVA Networks as Structured Implicit Layers and Vector Attractors via Dynamic Interaction Fields

arXiv: https://arxiv.org/abs/2607.28989 · HTML: https://mlsanity.com/paper/2607.28989/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 60 references that are in the map.
- First posted on arXiv 31 Jul 2026.

## Builds on (most cited first, 30 of 31)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Perceiver IO: A General Architecture for Structured Inputs & Outputs](https://mlsanity.com/paper/2107.14795/) (2021)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Principal Neighbourhood Aggregation for Graph Nets](https://mlsanity.com/paper/2004.05718/) (2020)
- [Residual Gated Graph ConvNets](https://mlsanity.com/paper/1711.07553/) (2017)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Multiscale Deep Equilibrium Models](https://mlsanity.com/paper/2006.08656/) (2020)
- [Implicit Graph Neural Networks](https://mlsanity.com/paper/2009.06211/) (2020)
- [A Clockwork RNN](https://mlsanity.com/paper/1402.3511/) (2014)
- [Stabilizing Equilibrium Models by Jacobian Regularization](https://mlsanity.com/paper/2106.14342/) (2021)
- [Attractor and integrator networks in the brain](https://mlsanity.com/paper/2112.03978/) (2021)
- [Lyapunov-Stable Deep Equilibrium Models](https://mlsanity.com/paper/2304.12707/) (2023)
- [Reversible Deep Equilibrium Models](https://mlsanity.com/paper/2509.12917/) (2025)

## Built on by (most cited first, 1 of 1)

- [Response Renormalization for Critical Deep Equilibrium Models](https://mlsanity.com/paper/2608.23725/) (2026)

## Abstract (arXiv)

Many learning problems require representations that reconcile direct input, nearby structure, and broader context. In implicit neural layers, these influences are usually absorbed into a single fixed-point update, making it hard to identify what enters from the stimulus, what propagates locally, what comes from global context, and what is produced by solver dynamics. Here we introduce SILVA Networks, Structured Implicit Layers and Vector Attractors via Dynamic Interaction Fields. SILVA separates stimulus, local interaction, global interaction, damping, and readout inside one fixed-point architecture. The same template is instantiated for images, molecules, citation networks, and long-range graph benchmarks through domain-specific definitions of nodes, neighborhoods, and global summaries. Experiments and ablations show task-dependent roles for these terms: local interactions are load-bearing in the graph tasks, MNIST gains little from recurrence at the tested capacity, and the clearest global benefit appears in a long-range node-classification benchmark. SILVA therefore provides an implicit representation whose internal interaction dynamics can be trained, ablated, visualized, and diagnosed.