# Finite Probes Suffice: Identifiability and Universality for Weight-Space Learning

arXiv: https://arxiv.org/abs/2609.33901 · HTML: https://mlsanity.com/paper/2609.33901/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 42 references that are in the map.
- First posted on arXiv 27 Sep 2026.

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

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Finding Neurons in a Haystack: Case Studies with Sparse Probing](https://mlsanity.com/paper/2305.01610/) (2023)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [On the equivalence between graph isomorphism testing and function approximation with GNNs](https://mlsanity.com/paper/1905.12560/) (2019)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Predicting Neural Network Accuracy from Weights](https://mlsanity.com/paper/2002.11448/) (2020)
- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [Permutation Equivariant Neural Functionals](https://mlsanity.com/paper/2302.14040/) (2023)
- [Graph Neural Networks for Learning Equivariant Representations of Neural Networks](https://mlsanity.com/paper/2403.12143/) (2024)
- [Graph Metanetworks for Processing Diverse Neural Architectures](https://mlsanity.com/paper/2312.04501/) (2023)
- [Neural Functional Transformers](https://mlsanity.com/paper/2305.13546/) (2023)
- [On the Optimal Memorization Power of ReLU Neural Networks](https://mlsanity.com/paper/2110.03187/) (2021)
- [Low Dimensional Invariant Embeddings for Universal Geometric Learning](https://mlsanity.com/paper/2205.02956/) (2022)
- [Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theorem](https://mlsanity.com/paper/2306.06529/) (2023)
- [Scale Equivariant Graph Metanetworks](https://mlsanity.com/paper/2406.10685/) (2024)
- [Equivariant Neural Functional Networks for Transformers](https://mlsanity.com/paper/2410.04209/) (2024)
- [Monomial Matrix Group Equivariant Neural Functional Networks](https://mlsanity.com/paper/2409.11697/) (2024)
- [Deep Linear Probe Generators for Weight Space Learning](https://mlsanity.com/paper/2410.10811/) (2024)
- [Learning Useful Representations of Recurrent Neural Network Weight Matrices](https://mlsanity.com/paper/2403.11998/) (2024)
- [Equivariant Polynomial Functional Networks](https://mlsanity.com/paper/2410.04213/) (2024)
- [Set-based Neural Network Encoding Without Weight Tying](https://mlsanity.com/paper/2305.16625/) (2023)
- [Analyzing Populations of Neural Networks via Dynamical Model Embedding](https://mlsanity.com/paper/2302.14078/) (2023)
- [Learning on Model Weights using Tree Experts](https://mlsanity.com/paper/2410.13569/) (2024)
- [On the Expressive Power of Permutation-Equivariant Weight-Space Networks](https://mlsanity.com/paper/2602.01083/) (2026)
- [Revisiting Multi-Permutation Equivariance through the Lens of Irreducible Representations](https://mlsanity.com/paper/2410.06665/) (2024)
- [What Linear Probes Miss: Multi-View Probing for Weight-Space Learning](https://mlsanity.com/paper/2605.23410/) (2026)
- [Provably Learning Attention with Queries](https://mlsanity.com/paper/2601.16873/) (2026)
- [Quasi-Equivariant Metanetworks](https://mlsanity.com/paper/2604.23720/) (2026)

## Abstract (arXiv)

Learning properties of neural networks has recently attracted growing interest, with existing approaches operating either directly on network parameters or through probe-based representations of network behavior. While probing methods have shown strong empirical performance, their theoretical foundations remain limited. In this work, we study when finite probe-based representations are sufficient for learning neural functionals. We establish general identification and universality results for probing, and show that using intermediate hidden representations can provide significantly more informative representations than relying only on final outputs. Motivated by these results, we introduce HIDDENPROBE, a simple architecture for learning from hidden probe responses. Across a range of neural functional benchmarks, including both MLPs and Transformers, HIDDENPROBE consistently improves over existing probing methods and achieves state-of-the-art performance. Our code is publicly available on GitHub.