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Fundamental Dynamical Units for Physics-Informed Structural Inference from Perturbation Time-Series in Networked Systems

Nima Nouri

arXiv:2609.11934v1cs.LGphysics.comp-ph

TL;DR

Recovering signed interactions from perturbation trajectories is difficult when distinct causal architectures produce indistinguishable responses. The paper uses FDU primitives, structure-derived perturbation design, and a physics-informed neural ODE; across five structural classes, the framework jointly recovers signed interactions and perturbation-resolved dynamics.

  • Problem

    Recovering signed interaction structure from trajectory observations is difficult because distinct causal architectures can produce indistinguishable responses without targeted interventions.

    Source: Introduction · p. 1

  • Method

    The framework combines signed three-node FDU primitives and structure-derived perturbation panels with a physics-informed neural ODE for joint interaction inference and trajectory prediction.

    Source: Introduction · p. 2, FDU-Regularized Continuous-Time… · p. 6

  • Results

    Across five structural classes and an added unary auto-inhibiting self-edge, the framework jointly recovers signed interactions and perturbation-resolved dynamics from sparse time courses without per-edge labels.

    Source: FDU-Regularized Physics-Informed… · p. 8

  • Takeaways & Limitations

    Each triad’s structural class prescribes perturbation conditions sufficient for signed-edge identifiability, allowing intervention design to be derived from the target motif.

    Source: Signed Triad Structure Dictates… · p. 5, Discussion · p. 10

    Validation is limited to synthetic data, and Hill kinetic parameters are fixed at ground-truth values rather than jointly inferred.

    Source: Discussion · p. 11

Abstract

from arXiv · show

In networked dynamical systems, the parameter of primary mechanistic interest is signed interaction structure. Recovering this structure from perturbation time-series data is a fundamental identification problem, compounded by three coupled obstacles: the combinatorial complexity of interaction architectures, ambiguity of causal attribution under limited interventions, and state-dependent dynamics that confound structural inference. Each obstacle is structural in origin and calls for a structural solution. We address these challenges by adopting a reductionist approach, introducing Fundamental Dynamical Units (FDUs): signed three-node interaction patterns as composable primitives that convert the interaction hypothesis space into a finite, constructive, and tractable representation. We show that local interaction structure determines the perturbation conditions required to disentangle direct from relayed influence, making intervention design a structural consequence of the FDU representation. We embed FDU-regularized structural inference within a physics-informed neural ordinary differential equation (ODE) whose governing-equation constraint transforms structural hypotheses into verifiable dynamical predictions, enabling joint recovery of interaction structure and perturbation-resolved trajectories. Validated on synthetic benchmarks with known ground truth, the framework supports structural commitment, expressed through FDU primitives, motif-prescribed intervention design, and physics-informed learning, as a principled basis for mechanistically interpretable inference in networked dynamical systems.

Introduction

Signed interaction recovery is structurally ambiguous under limited, sparse observations and state-dependent dynamics, so representation, intervention design, and dynamical learning must be addressed together. The paper proposes a FDU-regularized physics-informed neural ODE as a principled route to this coupled inference problem.

Introduction

  • Signed three-node Fundamental Dynamical Units (FDUs) make direct and relayed pathways explicit and provide compositional building blocks for tractable, mechanistically interpretable network architectures.A third node is the minimal addition that creates a parallel relay route and thus multi-path confounding.Source: Introduction · p. 1
  • Each local FDU specifies perturbation combinations that differentially excite parallel pathways, making interventions a structural consequence for distinguishing direct from relayed influence.These interventions are framed as targeted probes that make interaction hypotheses amenable to empirical refutation.Source: Introduction · p. 2
  • Operating-point dependence can hide an existing interaction, so continuous-time learning must track full trajectories, including transients, to separate invariant structure from state-dependent response amplitudes.At a single operating point, an absent edge may be indistinguishable from a present but locally insensitive one.Source: Introduction · p. 2
  • Unlike post-hoc motif vocabularies (13–15), signed FDU patterns define the hypothesis space and make structural attribution intrinsic to inference rather than ambiguous in overlapping subgraphs.Earlier directed-structure methods also treat perturbation panels as fixed rather than deriving them from local architecture, and leave operating-point dependence unresolved.Source: Introduction · p. 2
  • Standard neural ODEs, Neural CDEs, and physics-informed constraints alone do not jointly provide mechanistic interaction structure, causal attribution, and learning across state-dependent regimes.The paper positions its physics-informed neural ODE as coupling continuous-time learning with mechanistic constraints and an explicit interaction-structure representation.Source: Introduction · p. 2
  • The framework combines signed three-node structure, structure-derived perturbation design, and physics-informed continuous-time learning, demonstrated on synthetic benchmarks with known ground truth.The stated expectations are signed interaction recovery and motif-level attribution as falsifiable outcomes.Source: Introduction · pp. 2–3

Results

The Results establish and apply the FDU framework, defining signed-triad primitives, classifying direct–relay structure, deriving minimal perturbation panels from local edge signs, and embedding motif-based inference in a physics-informed ODE. Across synthetic motifs and network-embedded triads, they assess recovery of signed interactions and perturbation-resolved dynamics, including under open-system confounding, and show how FDU attention and latent forcing support inference.

The Signed Three-Node Interaction Space Admits an Exhaustive and Constructive Dictionary of Fundamental Dynamical Units

  • The FDU dictionary contains 70 motifs—64 signed tournaments and six unary self-motifs—and represents every sign-consistent pairwise-connected triad using at most two tournament FDUs.Triads without mutual pairs need one tournament FDU; those with mutual pairs need two, with multiple minimal decompositions treated as representational redundancy.Source: The Signed Three-Node Interaction… · pp. 3–4

The Signed Three-Node Interaction Space Partitions into Five Permutation-Invariant Structural Classes

  • The permutation-invariant taxonomy assigns each triad to one of five classes; 496 of 512 (96.9%) are parallel-path motifs, while 16 (3.1%) are single-path.Among parallel-path motifs, 248 are mixed-coherence, 124 coherent, and 124 incoherent; the 16 single-path motifs split evenly between positive and negative feedback.Source: The Signed Three-Node Interaction… · p. 4

Signed Triad Structure Dictates Minimal Perturbation Panels for Relay Cancellation

  • Relay-cancellation panel polarity depends only on the first relay-leg sign: activating edges require opposite-sign co-perturbation, while inhibiting edges require same-sign co-perturbation.Single-component perturbations suffice for single-path motifs; mixed-coherence motifs require both co-perturbation polarities to support attribution across parallel pairs.Source: Signed Triad Structure Dictates… · p. 5
  • 290 of 512 (57%) admissible triads require dual-polarity co-perturbation panels, the most common panel requirement across all signed triads.All 248 mixed-coherence motifs require dual polarity; the panel assignments assume local linear response and matched perturbation amplitudes.Source: Signed Triad Structure Dictates… · pp. 5–6

FDU-Regularized Continuous-Time Dynamics Constitute a Constructive Physics-Informed Framework for Signed Motif Inference

  • FDU attention constrains signed topology while separate strength parameters encode interaction magnitudes, and ODE consistency across prescribed perturbation panels narrows structures that fit observed trajectories.The finite FDU bank grows as O(N^3); physics-informed training requires trajectories to satisfy the governing ODE at observed time points.Source: FDU-Regularized Continuous-Time… · p. 6
  • The continuous-time framework uses saturating, state-dependent interactions and channel exclusivity so that interaction signs are encoded by the dynamics and the three coupled inference challenges are addressed structurally.Trajectory phases near the Hill response’s half-saturation scale carry discriminative signal; under channel exclusivity, the sign of the local derivative encodes the signed edge.Source: FDU-Regularized Continuous-Time… · p. 7

FDU-Regularized Physics-Informed Inference Recovers Signed Structure, Motif Identity, and Dynamics Without Per-Edge Structural Labels

  • Across six motifs, activating and inhibiting channels recovered ground-truth edges and polarity while suppressing non-edges, with true-edge parameters selectively stabilizing in the appropriate channel.The signed partition emerged from training rather than per-edge supervision.Source: FDU-Regularized Physics-Informed… · pp. 7–8
  • Across six motifs, predicted trajectories tracked ground truth over the full horizon, with median range-normalised root-mean-square error from 0.12 to 0.21 across motifs.The test set covered held-out time points across conditions and states; the results indicate fidelity across perturbation panels from sparse multi-phase supervision.Source: FDU-Regularized Physics-Informed… · p. 7
  • FDU attention localized to generating motif support: five motifs selected a single FDU, while motif 000201220 used its minimal two-FDU decomposition.For the composite motif 000102102, inference recovered an auto-inhibiting self-edge and localized attention to its triadic and unary components, with median range-normalised root-mean-square error of 0.19.Source: FDU-Regularized Physics-Informed… · pp. 7–8

FDU-Regularized Inference Recovers Signed Interaction Structure of a Network-Embedded Triad Under Open-System Confounding

  • In open systems, latent forcing represents the environmental remainder as a smooth, time-varying, condition-specific term distinct from persistent FDU-structured interactions, enabling joint estimation of forcing and signed structure.Across embedded triads, the results show that open-system confounding can be separated rather than treated as a fundamental barrier to signed structural inference.Source: FDU-Regularized Inference Recovers… · pp. 8–10
  • In the 20-node network, all seven directed edges from the target triad were recovered with correct polarity, with median range-normalised root-mean-square error of 0.12.The two edges from node 1 were activating, node 16’s three edges were inhibiting, and node 18 had one activating and one inhibiting edge.Source: FDU-Regularized Inference Recovers… · p. 9
  • Of 73,001 candidate placements in the embedded-triad analysis, 72,995 had zero or near-zero attention weight, while top-ranked FDUs included the target decomposition and confirmed neighboring interactions.The target triad’s minimal two-FDU decompositions appeared within the top ten; higher-weight FDUs beyond that decomposition involved nodes 4 and 5, the most responsive non-target nodes.Source: FDU-Regularized Inference Recovers… · pp. 9–10
  • Across ten embedded triads, AUROC is 0.998 and F1 = 0.93; disabling latent forcing reduces AUROC = 0.60 and F1 = 0.27.The five false negatives were intra-triad edges; outgoing-edge sensitivity was perfect across all ten triads and four panel types.Source: FDU-Regularized Inference Recovers… · p. 10

Discussion

The discussion frames signed-structure inference as a representational problem: FDU primitives align what data identify with what the model represents, while linking structural hypotheses to perturbation design and dynamical predictions.

Discussion

  • FDU reframes signed-structure inference as a representational rather than data problem by aligning the hypothesis space, perturbation design, and dynamical model around the same primitive.Under this construction, the data and model agree on signed structure rather than requiring it to be extracted from data alone.Source: Discussion · pp. 10–11
  • The triad is the minimal closed unit where direct and relay pathways coexist; the FDU dictionary represents every signed triad using at most two tournament primitives and scales as O(N^3).Its primitives couple edge choices within three-node subgraphs, making pathway composition intrinsic; the dictionary is interpretable and self-specifies perturbation conditions sufficient for signed-edge identifiability.Source: Discussion · p. 10
  • Every recovered activating or inhibiting edge is attributed to supporting FDU primitives, whose attention weights quantify contributions and separate relay-mediated from direct influence (cf.40).This attribution is built into the hypothesis space rather than applied after recovery, including when multiple subgraphs overlap.Source: Discussion · p. 10
  • A target triad’s signed structural class prescribes the minimal perturbations needed to cancel relay effects and isolate direct arms, preventing conflation that can create spurious direct edges.When the local class is unknown, single-component perturbations combined with dual-polarity co-perturbations cover all admissible signed triads.Source: Discussion · pp. 10–11
  • Validation is limited to synthetic data with ground-truth Hill kinetic parameters, leaving real-world robustness and joint kinetic-parameter estimation untested.The O(N^3) bank can make full-bank attention costly at larger scales; entmax sparsification concentrates attribution on compact active support.Source: Discussion · p. 11

Methods

The method jointly predicts perturbation-resolved trajectories, infers sparse FDU interaction structure, and absorbs dynamics outside the modeled subnetwork. Its three modules are trained under a composite objective and progressive optimization schedule.

Model Architecture

  • A reference-conditioned predictor uses the unperturbed state at each query time and explicit perturbation input to predict trajectories across conditions and focus learning on perturbation-driven deviations.A tanh multilayer perceptron supplies exact time derivatives through forward-mode automatic differentiation, including on irregular time grids.Source: Model Architecture · p. 11
  • Entmax-regularized attention selects sparse FDU support, while separate nonnegative strength matrices estimate interaction magnitudes.Topology masks gate the strengths; motif logits start at zero, giving a uniform prior over candidate placements.Source: Model Architecture · pp. 11–12
  • A condition-dependent latent forcing module models contributions the FDU structure cannot represent, including influences from outside the modeled subnetwork.It combines perturbation-dependent coefficients with an analytic Fourier time basis; the rank of its node-mixing matrix controls environmental-drive complexity.Source: Model Architecture · p. 12

Training Objective

  • The composite objective balances an ODE physics residual with trajectory fidelity, initial-state anchoring, non-negativity, and activation-inhibition exclusivity.Uncertainty-based adaptive weights apply to these five terms; separate fixed-weight penalties encourage forcing that is small, smooth, sparse in node support, and not aligned with the perturbation channel.Source: Training Objective · p. 12

Optimization Protocol

  • Joint Adam optimization anneals entmax concentration from the softmax limit toward near-sparsemax using a cosine schedule, then uses validation-based learning-rate reduction, gradient clipping, and early stopping.The schedule begins slowly, accelerates as perturbation responses disambiguate FDU support, and decelerates near its target to ease commitment.Source: Optimization Protocol · pp. 12–13

Simulation and Experimental Setup

  • Synthetic trajectories use sparse samples from four fixed time windows, with held-out test points across the full horizon, while perturbation panels use signed single-node and co-perturbation conditions.The 20-node network comprises 10 randomly placed signed triads; the latent forcing module is enabled for this experiment and disabled in triad-only experiments.Source: Simulation and Experimental Setup · p. 13

Computational Resources

  • The 20-node FDU bank had 73,000 candidate placements versus 70 for three nodes, yet training time increased by less than twofold, attributed to entmax sparsification.Experiments ran on a CPU-only platform.Source: Computational Resources · p. 13

Data Availability

The study used only computationally generated synthetic datasets; no empirical data were collected or used.

Data Availability

  • Synthetic datasets were generated using code at https://github.com/AstraZeneca/fdu-pisi, and the datasets generated and/or analysed are available at Zenodo DOI 10.5281/zenodo.20377559.No empirical data were collected or used.Source: Data Availability · p. 13

Competing Interests

The author reports being an employee of AstraZeneca US.

Competing Interests

f c e

FDU-regularized inference is presented as recovering signed interaction structure and motif identity under open-system confounding. Minimal perturbation panels are assigned according to triad structure and relay-cancellation requirements.

  • In a 20-node network with open-system confounding, FDU-regularized inference recovers signed interactions and the P_in-c motif among target nodes 1, 16, and 18.The figure also shows held-out-time-point error across 13 perturbation conditions and FDU attention weights for target-triad components and surrounding-network motifs.Source: Figure 6 · p. 22
  • For signed triads without parallel ordered pairs (class S), single-node perturbations suffice to cancel relayed contributions and support attribution.These triads are assigned to the (u, 0) category under Rule 1.Source: Table 1 · p. 24
  • For triads with parallel ordered pairs, panels require one co-perturbation polarity or both when relay-cancellation requirements conflict or coherence is mixed.Rule 2 assigns one polarity based on the first relay-leg sign; conflicting requirements or mixed coherence in class P_x require both polarities.Source: Table 1 · p. 24

Description of Additional Supplementary Files

The supplementary files provide a catalog of signed three-node triads and additional figures, a table, and a note.

Description of Additional Supplementary Files

  • The catalog covers all 512 pairwise-connected signed triads, with structural classes, minimal perturbation panels, and compatible FDU decompositions.Rows also report each triad’s motif code, mutual-pair count, and edge count.Source: Description of Additional… · p. 25
  • The Supplementary Information contains three figures, one table, and a supplementary note.Source: Description of Additional… · p. 25

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