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

Kaiming Luo

cs.LGDec 6, 2025

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

Authors: Kaiming Luo

Organizations: School of Information Science and Technology, Fudan University, Shanghai, 200438, China

Abstract

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.

Figures & tables

Figure 1: Overview of the VPIA steady-state reconstruction framework. (A) Representative planted interaction structure used only for validation. The framework is tested on undirected, directed, weighted, and selected higher-order interaction patterns. The light-blue shaded region indicates a representative 2-simplex. (B) Collection of heterogeneous steady-state observations generated under different intrinsic parameters or driving conditions. Each trial $m$ relaxes to a stable configuration $x^{*(m)}$ , which is used for reconstruction. (C) Variational representation and equilibrium-constrained optimization. The reconstruction module receives the observed steady states, known controls, and governing equations, without access to the true interaction structure. Each candidate interaction is assigned a trainable parameter and mapped to a continuous admissibility variable through a differentiable ansatz. (D) Reconstructed interaction operator $\widehat{A}$ obtained by minimizing the steady-state residuals across experiments. (E) Quantitative evaluation of $\widehat{A}$ against the ground-truth structure $A$ , with AUC used as the primary reconstruction metric.

Figure 2: Reconstruction performance of the VPIA framework. (A) Ground-truth adjacency matrix, with black and white entries denoting the presence and absence of links. (B) ROC curve for Kuramoto steady-state data. The upper-left inset reports AUC for the indicated dynamical models, and the lower-right inset shows AUC versus the number of experiments. (C) Structural-recovery experiment counts versus network size, averaged over 20 realizations with standard-deviation bands; recovery requires AUC $\geq 0.999$ and zero support errors at threshold $0.5$ . The inset reports accuracy, precision, recall, F1 score, and AUC. No added noise is used.

Figure 3: Synthetic robustness tests under three classes of zero-mean noise. (A) Observation noise, in which each measured steady state is contaminated by additive random fluctuations. (B) Dynamical noise, modeled as a stochastic term acting during the time evolution. (C) Structural noise, corresponding to perturbations of the underlying interaction coefficients. For each noise class, three representative amplitudes are shown in the right-hand panels. Across all cases, reconstruction accuracy improves as the number of heterogeneous steady-state experiments increases. For structural noise, increasing $M$ suppresses sampling fluctuations but does not remove the perturbation-induced accuracy floor relative to the clean underlying structure. All curves are averaged over 20 independent realizations; $M$ varies along the horizontal axis in each panel, $N$ is fixed within each panel, and the noise amplitudes are $0.1$ , $0.3$ , and $0.5$ , as indicated in the legends.

Figure 4: Accuracy comparison on an ER benchmark. Mean AUC for VPIA, ND, PPS, MRA, GS, and Corr at $N=50$ . Bars and error bars show means and standard deviations over 20 realizations.

Operation	Work
Sampled residuals	$O(Msk_{c})$
Dense parameter update	$O(E_{c})$
Iterative inference	$O[T(Msk_{c}+E_{c})]$
All-pairs, full residuals	$O(MN^{2})$ per iteration
Dense $N\times N$ inversion	$O(N^{3})$ per solve

Table 1: Dominant pairwise work for explicitly specified computational regimes.

T

is the optimization iteration count. Candidate degree

k_{c}

is distinct from true degree.

Network	Type	Nodes	Edges	Density
Florentine Families	Undir	16	20	0.1667
Zachary Karate Club	Undir	34	78	0.1390
Wild Bird Social	Undir	202	11733	0.5861
C. elegans Neural	Dir	297	2359	0.0259
Power	Undir	494	596	0.0049
Retweet Weibo	Dir	596	1415	0.0080

Table 2: Benchmark empirical graph topologies analyzed in this work. Publicly available graph structures are used as fixed ground-truth interaction topologies for synthetic steady-state validation. For each network, we report its type, number of nodes

N

, number of edges, and density

\rho

Figure 5: Empirical-topology synthetic benchmarks. Upper panels: representative benchmark graphs used as fixed planted interaction structures for synthetic steady-state tests, ranging from small social networks to directed communication networks and a large connectome-like graph. Lower panels: AUC as a function of the number of heterogeneous steady-state experiments for synthetic steady states generated on the same benchmark topologies. Insets show the corresponding degree or in-degree distributions. The empirical graphs serve as fixed planted interaction structures for synthetic steady states generated under known dynamics. The suite spans $N=16,34,202,297,494,596,$ and $989$ across the seven panels, with $M$ varied along the horizontal axes and no added noise.

Figure 6: Higher-order reconstruction. (A) Illustrations of $1$ -, $2$ -, $3$ -, and $4$ -simplex interaction motifs, corresponding to two-, three-, four-, and five-vertex interaction structures. (B) Representative reconstruction of a $2$ -simplex adjacency tensor. The left panel shows the planted higher-order tensor, and the right panel shows the tensor recovered by VPIA. (C) Reconstruction accuracy, measured by AUC, as a function of the number of steady-state experiments. (D) Number of steady-state experiments required for structural recovery across systems composed of $1$ -, $2$ -, $3$ -, and $4$ -simplex interactions. Higher-order simplices require more experiments because the number of unknown interaction coefficients grows combinatorially with simplex order. (E) Robustness under observation noise (left), dynamical noise (middle), and structural noise (right). Although larger noise amplitudes reduce the initial AUC, discrimination improves systematically as additional heterogeneous steady-state constraints are incorporated. These tests use dense higher-order systems of the sizes specified below. Panels (C)–(E) use networks with $N=10$ , where all possible $2$ -, $3$ -, and $4$ -simplex interactions are independently included with probability $p=0.5$ . Panels C and E use $N=10$ , panel D scans $N=5$ to $20$ , $M$ is varied along the horizontal axes in panels C–E, and the noise amplitudes in panel E are $0.01$ , $0.05$ , and $0.1$ .

Figure S1: Reconstruction across different network topologies. Illustration of true versus reconstructed interaction structures for three different network topologies: (a) Barabási–Albert (BA) network with $N=100$ , $m=2$ , (b) Watts–Strogatz small-world (SW) network with $N=100$ , $k=4$ , $\beta=0.1$ , and (c) Erdős–Rényi (ER) network with $N=100$ , $p=0.05$ . In all three topologies, VPIA recovers the main edge patterns, including hub nodes in BA networks, shortcut connections in SW networks, and random connectivity in ER networks. These examples are consistent with recovery across the tested structural ensembles.

Figure S2: Directed network reconstruction. (a) Example directed network with $N=50$ . (b) Comparison between true and reconstructed in-/out-degree distributions, showing accurate capture of directed edge asymmetries. (c) AUC for directed edge prediction over $N=50$ – $100$ (averaged over 20 independent runs), consistently achieving high performance between $0.99$ and $1.00$ .

Figure S3: Weighted network reconstruction. (a) Example weighted network with $N=50$ and integer weights. (b) True and reconstructed degree distributions, showing that VPIA captures the degree structure accurately. (c) Numerical weight error $\delta$ versus network size $N$ , with errors consistently below $2\%$ . (d) AUC for binary support prediction, demonstrating high recovery accuracy with AUC values between $0.98$ and $1.00$ .

Method	Main computational steps	Time complexity
VPIA	Sample $r$ residuals; Fisher-preconditioned update.	$\mathcal{O}(r\rho N)$
Steady-state Jacobian regression (SJR)	$N$ high-dimensional regressions.	$\mathcal{O}(MN^{2}+N^{3})$
Local regression from dynamics	Node-wise nonlinear regression across $M$ states.	$\mathcal{O}(MN^{2})$
Partial Phase Synchronization (PPS)	Construct and invert synchronization matrix.	$\mathcal{O}(N^{3})$
Network Deconvolution (ND)	Eigen-decomposition of interaction matrix.	$\mathcal{O}(N^{3})$
Modular Response Analysis (MRA)	Inversion of global response matrices.	$\mathcal{O}(N^{3})$

Table S1: Computational complexity of representative network reconstruction methods.

N

: number of nodes;

M

: number of steady states;

\rho

: network density;

r

: number of sampled node–state residuals per iteration.

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