# Variational Physics-Informed Ansatz for Reconstructing Hidden Interaction Networks from Steady States

arXiv: https://arxiv.org/abs/2512.13708 · HTML: https://mlsanity.com/paper/2512.13708/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 37 references that are in the map.
- First posted on arXiv 6 Dec 2025.

## Builds on (most cited first, 1 of 1)

- [Network reconstruction via the minimum description length principle](https://mlsanity.com/paper/2405.01015/) (2024)

## Abstract (arXiv)

Inferring interaction structure from steady-state observations is a central inverse problem when transient trajectories are unavailable. Here we formulate this problem as simultaneous compatibility of a single interaction operator with equilibrium constraints generated by heterogeneous perturbations. We introduce a variational physics-informed ansatz that represents the unknown operator as a trainable object and minimizes the resulting steady-state residuals across experiments. In the affine-interaction setting, the stacked equilibrium equations yield explicit finite-sample identifiability conditions: unique recovery is controlled by the rank of the compatibility matrix after elimination of experiment-wise gauge freedom. Synthetic benchmarks on pairwise, directed, weighted, empirical-topology, and selected higher-order systems illustrate this identifiability picture and show how additional heterogeneous steady states improve structural discrimination under the stated assumptions. The results clarify a concrete steady-state reconstruction regime in which equilibrium observations alone can determine hidden interaction operators when the governing dynamics are known and node-level equilibria are fully observed.