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Fundamental Dynamical Units for Physics-Informed Structural Inference from Perturbation Time-Series in Networked Systems
Nima Nouri
TL;DR
Signed interaction structure is difficult to identify because network architectures are combinatorial, interventions are limited, and observable effects depend on operating state. The paper introduces FDU primitives and embeds them in a physics-informed neural ODE with structure-derived perturbation panels. On synthetic benchmarks, the framework recovers signed structure, motif identity, and perturbation-resolved dynamics, including AUROC 0.998 across embedded triads.
Problem
Signed interaction structure remains under-constrained by trajectory data because direct and relayed effects can be indistinguishable, especially under limited interventions and state-dependent dynamics.
Method
The framework represents signed triads with Fundamental Dynamical Units, derives perturbation panels from local structure, and constrains a neural ODE with physics-informed dynamics.
Results
The framework jointly recovers signed interactions, generating motifs, and perturbation-resolved dynamics without per-edge structural labels; pooled triad scoring achieves AUROC 0.998 and F1 = 0.93.
Takeaways & Limitations
FDU structure provides a finite, interpretable, and tractable hypothesis space whose motif class prescribes perturbation conditions for signed edge identifiability.
Takeaways & Limitations
Validation is synthetic, kinetic parameters are fixed at ground-truth values, and robustness to real-world acquisition, noise, misspecification, and preprocessing remains unestablished.
Abstract
from arXiv · showhide
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 a structural identification problem because combinatorial architectures, limited interventions, and state-dependent dynamics make distinct mechanisms observationally indistinguishable. The framework addresses these constraints with FDU primitives, structure-derived perturbation design, and a physics-informed neural ODE for joint mechanistic inference.
- Problem: Signed interaction recovery is non-injective: distinct causal architectures can produce indistinguishable trajectories under partial measurements, limited interventions, and state-dependent dynamics.More data of the same design or more expressive models do not generally resolve this structural ambiguity.
- Design principle: The framework treats representation, causal attribution, and dynamical learning as coupled design constraints rather than independent modeling choices.This links the interaction hypothesis space to intervention requirements and to models that preserve mechanistic meaning across operating regimes.
- Structural representation: Fundamental Dynamical Units represent signed three-node patterns as composable primitives, matching the minimal triad structure that creates direct-versus-relayed pathway ambiguity.FDUs provide a local basis for assembling network-scale architectures while making parallel pathways explicit.
- Interpretability: The approach makes structural attribution intrinsic by identifying which signed primitives generate recovered interactions and their compositional weights.This supports compact, testable hypotheses about multi-component dynamical systems from heterogeneous observational and interventional time-series data.
- Framework: A FDU-regularized physics-informed neural ODE jointly predicts trajectories and infers signed interactions from synthetic benchmarks with known ground truth.Its contributions include exhaustive signed-pattern encoding, structure-derived perturbation design, and continuous-time learning with explicit structural priors.
Results
The FDU dictionary provides a complete, compositional representation of signed three-node architectures, while structural classes determine minimal perturbation panels for separating direct and relay pathways. Embedded in a physics-informed model, this representation scales polynomially and converts local structure into actionable intervention design.
- The FDU dictionary contains 70 motifs: 64 triadic tournaments and 6 unary self-motifs for auto-regulation.The triadic tournament count is 4^3 = 64, with three auto-activating and three auto-inhibiting self-motifs.
- Every admissible signed three-node configuration is exactly recoverable from at most two tournament FDUs under sign-consistent union.A single tournament FDU suffices for triads without mutual pairs; two resolve mutual pairs while preserving forced edges.
- Parallel direct-relay structure dominates the admissible space, comprising 496 of 512 triads (96.9%), while only 16 are single-path motifs.The parallel class further separates into coherent, incoherent, and mixed subclasses according to sign coherence across parallel ordered pairs.
- Minimal intervention panels are defined under local linear-response and matched-amplitude assumptions to cancel relayed contributions across all parallel ordered pairs.The four panel categories are single-component, opposite-sign only, same-sign only, and dual-polarity; the assignment is determined by signed triad structure.
- 290 of 512 admissible signed triads (57%) require dual-polarity co-perturbation panels, making this the most common intervention requirement.Across parallel-path motifs, 103 require only opposite-sign, 103 only same-sign, and 290 dual-polarity co-perturbations.
- Structural commitment is implemented through a complete FDU bank, attention over primitives, topology-magnitude factorization, and physics-informed operating-point-aware dynamics.The hypothesis space grows as O(N^3), with L(3) = 70, L(4) = 264, and L(20) = 73,000.
Discussion
The FDU framework aligns the structural level of representation with what perturbation data can identify, using signed triads as complete, interpretable, self-specifying, and tractable primitives. It derives perturbation panels from local structure to separate direct and relay-mediated effects, while remaining bounded by synthetic validation and fixed kinetic parameters.
- Discussion: Signed triads align the model’s parameterization with the structural level that trajectory data can identify.The framework treats triads, rather than independent edges, as carrying the mechanistic content of signed interactions.
- Discussion: The FDU dictionary represents every admissible signed triad using at most two tournament primitives and scales as O(N^3).It is also interpretable through explicit pathway and perturbation-design consequences and self-specifying because structural classes prescribe identifiability conditions.
- Discussion: FDU superposition couples edge choices within three-node subgraphs, making pathway composition and coherence intrinsic rather than inferred from independent adjacency entries.Complex architectures are compositions of the same primitive set rather than bespoke representations.
- Discussion: FDU support attributes every learned activating or inhibiting edge to specific primitives, separating relay-mediated from direct contributions at the structural level.Attention weights quantify each primitive’s contribution to the observed dynamics.
- Discussion: Triad classes prescribe perturbation conditions for relay cancellation and direct-arm isolation, while single-component and dual-polarity co-perturbations cover unknown local structures.This makes intervention design a class-to-panel consequence rather than an independent choice.
- Discussion: Validation remains limited to synthetic data with ground-truth kinetic parameters, leaving robustness to real-world variation and joint kinetic-structure estimation unresolved.The O(N^3) bank also becomes computationally non-trivial at larger network scales, although entmax concentrates attribution on active support.
Methods
The method jointly predicts perturbation trajectories, infers sparse FDU-supported structure, and absorbs unrepresented influences within a physics-informed neural ODE. Reference conditioning, differentiable structural gating, adaptive losses, and progressive sparsification preserve perturbation-response information while enforcing mechanistic constraints.
- Methods: Three jointly trained modules predict trajectories, parameterize sparse FDU structure, and absorb contributions outside the modeled subnetwork.A composite objective and progressive optimization schedule preserve the perturbation-response signal needed for structural inference.
- Methods: Conditioning on the unperturbed reference state focuses the predictor on perturbation-driven deviations rather than relearning baseline dynamics.The predictor maps time, reference state, and perturbation vector to the predicted state.
- Methods: Entmax attention over the FDU bank separates topology from interaction magnitude and lets sparse structural support emerge from optimization.Zero-initialized logits impose a uniform prior over candidate placements, while fixed Hill parameters restrict inference to effective interaction strengths.
- Methods: The latent forcing module represents unmodeled, condition-dependent dynamics with a smooth finite-dimensional Fourier basis and a perturbation-conditioned coefficient network.A learnable node-mixing matrix combines the basis functions into the forcing signal.
- Methods: The physics residual differentiates through FDU-induced support masks and the ODE right-hand side, coupling structural attention to dynamical consistency.Additional terms enforce data fidelity, initial-condition anchoring, non-negativity, and channel exclusivity.
- Methods: Uncertainty weighting adapts the five main loss terms, with physics and data fidelity initially weighted more strongly than auxiliary constraints.The initial effective weights for physics and data are e^2 ≈ 7.4, versus 1 for initialization, non-negativity, and exclusivity.
- Methods: Joint optimization uses entmax annealing, bounded log-variances, validation-based learning-rate reduction, and gradient clipping to stabilize structural commitment.Annealing moves from broad softmax exploration toward near-sparse support rather than imposing sparsity abruptly.
Data Availability
The study uses computationally generated synthetic datasets and provides the generation code and analyzed datasets through public repositories.
- Data Availability: All datasets are synthetic and computationally generated; no empirical data were collected or used.Code is available on GitHub, and datasets are deposited in Zenodo.
Competing Interests
The author reports employment by AstraZeneca US as a competing interest.
- Competing Interests: The author is an employee of AstraZeneca US.
Surface Triangulation Volumetric Triangulation
The framework replaces boundary-only pairwise connectivity with volumetric FDU primitives that encode direct and relayed pathways, then uses these structures to guide interventions and physics-informed dynamical inference.
- Surface Triangulation: Surface triangulation encodes boundary connectivity, whereas volumetric triangulation represents networks through interior-filling signed three-node FDU primitives.FDUs constrain how edges compose into multi-path structures rather than representing only pairwise connectivity.
- Volumetric Triangulation: FDU dictionaries classify signed triads and decompose composite structures through superposition, including mutual interactions and self-regulation.The dictionary supports permutation-invariant structural classes and constructive decompositions of signed triadic architectures.
- Volumetric Triangulation: Minimal perturbation panels are prescribed by structural class to cancel relayed contributions and isolate direct interactions.Single-path motifs use single-node perturbations, while parallel-path motifs require polarity-specific or dual-polarity co-perturbations.
- Volumetric Triangulation: FDU embedding combines sparse entmax attention over motif primitives with separate activating and inhibiting channels for topology and interaction magnitude.Hill-type signed dynamics preserve correspondence between learned channels and activating or inhibiting interactions.
- Volumetric Triangulation: Across representative motifs, the framework recovers signed edges, motif identity, and held-out trajectories under class-specific perturbation conditions.Structural recovery is read from the activating and inhibiting channels, while attention identifies single-FDU or compositional representations.