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

Models of complex systems often have many parameters, yet are constrained by far fewer experimentally accessible observables: similar activity can emerge from coordinated parameter changes.

We formalize these compatible parameter sets as \emph{viable parameter manifolds}: the inverse images of a system's target dynamical behaviors 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.

Co-varying features lower the 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 two-feature 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 recent ODE reduction of finite spiking networks, the same framework reveals excitatory--inhibitory compensation, timescale--coupling tradeoffs, and input-dependent 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.