# Diffusion learning reveals viable parameter manifolds and compensation geometry in biological dynamical systems

arXiv: https://arxiv.org/abs/2607.03671 · HTML: https://mlsanity.com/paper/2607.03671/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 6 of its 41 references that are in the map.
- First posted on arXiv 4 Jul 2026.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 6 of 6)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Dimensionality reduction of neuronal degeneracy reveals two interfering physiological mechanisms](https://mlsanity.com/paper/2405.02038/) (2024)
- [Minimizing information loss reduces spiking neuronal networks to differential equations](https://mlsanity.com/paper/2411.14801/) (2024)
- [A data-informed mean-field approach to mapping of cortical parameter landscapes](https://mlsanity.com/paper/2110.12286/) (2021)
- [Multi-band oscillations emerge from a simple spiking network](https://mlsanity.com/paper/2206.14942/) (2022)

## Abstract (arXiv)

Models of complex systems often have many parameters, yet are constrained by far fewer experimentally accessible observables; consequently, similar activity can emerge from coordinated parameter changes. We formalize these compatible parameter sets as viable parameter manifolds: the inverse images of target dynamical features under a parameter-to-feature map. The relevant codimension is not the number of reported features, but the effective rank of that map at the target scale. Locally redundant features lower the effective codimension, while poor conditioning, high curvature, or regime mixing degrade learnability. We train conditional score-based diffusion models on simulated parameter--feature pairs and use them as amortized samplers of prior-weighted viable sets. In the Lorenz system, scalar trajectory statistics generate thin viable sheets, and a finite-tolerance conditioning localizes a transition-adjacent corridor. In the Izhikevich neuron model, four firing descriptors lie close to a nearly two-dimensional family of features, and the learned inverse images reveal distinct regular and irregular compensation geometries. In a deterministic ODE reduction of finite spiking networks, the same framework reveals excitatory--inhibitory compensation, timescale--coupling tradeoffs, and viable manifolds across 4--12 parameter dimensions. In this view, robustness, compensation, and hidden parameter dependencies are organized as inverse geometry, with diffusion models providing practical tools for sampling, visualizing, and interrogating that geometry.