Edge Accuracy Is Not Enough: Why Dynamics-Learned Structure Fails to Transfer to Inverse Problems

The gist: Structure learned via Neural Relational Inference (NRI) from dynamics prediction satisfies the edge-accuracy condition, yet it underperforms even a fixed, physics-agnostic baseline because approximation error dominates when forward and inverse tasks have different optimal structures.

Theoretical Foundation

The paper establishes a theoretical starting point by showing that known sparse structure reduces sample complexity from O(n 2) to O(kn) (Theorem 1), and this benefit degrades gracefully under edge-level errors as long as ∆ < n2 − kn (Theorem 3). The core finding is that the edge-accuracy condition bounds estimation error, not approximation error, which is what dominates when the forward and inverse tasks have different optimal structures

Empirical Failure of Structure Transfer

The study systematically demonstrates that structure learned via NRI from forward dynamics fails to transfer to source-localisation tasks across two physically distinct domains: CFD simulations of hydrogen leaks and acoustic scenarios On the source-localisation task, NRI degrades performance by 116% in the CFD domain and 201% in the acoustic domain relative to a flexible, task-optimised attention baseline

Four Lines of Evidence Against Tuning Failure

The failure is not attributed to insufficient data or poor hyperparameter choices through four independent lines of evidence

Evidence 1: Data Scaling shows a flat scaling curve under increasing data, which is the signature of approximation error, not estimation error

Evidence 2: Threshold Insensitivity shows that performance remains identical across a wide range of graph density thresholds τ ∈ [0.2, 0.5], ruling out a simple sparsity-tuning explanation

Evidence 3: Structural Overlap shows the dynamics-learned graph overlaps the task-optimal graph on only 6% of edges (Jaccard similarity), which is far short of the threshold required for successful transfer

Evidence 4: The failure is not NRI-specific, as other dynamics-derived structure estimators, such as correlation- and mutual-information-based estimators, show no measurable benefit over a structure-free baseline

Practical Transferability Test

The authors propose a practical transferability test based on Jaccard overlap between the candidate structure and a partially-observed target-task graph The procedure involves holding out 15–20% of target-task data, training a minimal localiser, extracting its task-specific attention graph GˆB, and rejecting transfer if J(G, ˜ GˆB) < 0.30

Conclusion and Guidance

The paper concludes that the benefit of structure transfer is contingent upon the learned structure being task-relevant, as forward dynamics optimize for temporal coherence while inverse localisation optimizes for spatial constraint Practical guidance suggests using physics-based structure when reliable domain knowledge constrains the inverse task directly and using flexible, task-specific learning when accuracy is paramount The final recommendation is to only attempt structure transfer when J ≥ 0.30 against a partially observed target-task graph

AI Use Statement

In this work, we used generative AI tools to draft initial implementations of the localiser baselines, structure estimators, and simulation pipelines; to assist with post-hoc debugging of the experiment pipeline, including identifying a methodological issue in how one baseline’s evaluation was constructed; to draft interpretive prose describing what the results show; to draft and edit portions of the manuscript; and to source and summarise related literature All AI-assisted code, debugging, interpretation, and literature suggestions were independently verified by the authors against the underlying data and original sources before inclusion, including independently re-running experiments to confirm the results in Tables 1–8 We did not use generative AI to develop the theoretical models, formulate or prove the mathematical claims (Theorems 1–3, Conjecture 1), propose or refine hypotheses, design the research methodology or experiments, generate or clean data, support qualitative data analysis, create or modify figures, suggest experimental parameters, or propose the paper’s structure or title; these and the final interpretation of all findings remain the authors’ own responsibility

Reproducibility Statement

Code, NRI checkpoints, and experiment scripts are available at https://github.com/nicolaisi/edge-accuracy-is-not-enough CFD scenario data are available on Zenodo in four parts (part 1, part 2, part 3, part 4); the acoustic domain is fully reproducible from a provided generator script (under 15 seconds per full 180-scenario sweep on a single CPU core) Full proofs of all theoretical results are in Appendix A; complete architectural, training, and dataset details are in Appendices B–C

Ethics Statement

This work supports hydrogen safety monitoring and acoustic source localisation We do not identify a path by which our specific contributions (sample-complexity theory and a structure-transferability diagnostic) facilitate harmful surveillance beyond what is already enabled by source-localisation methodology in general use

References

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1. Improve structure transfer reliability by implementing a practical transferability test: "train a minimal localiser on it; extract its task-specific attention graph GˆB; compute J(G, ˜ GˆB) for the candidate structure G˜; reject transfer if J < 0.30." This diagnostic separates successful from failed transfer in all four domain/structure pairs evaluated.

2. Develop a robust thresholding mechanism by avoiding reliance on single-point calibration: "The wide plateau across τ ∈ [0.2, 0.5], the negligible difference at the F1-optimal threshold (τ = 0.1, within 0.1% relative of the plateau), and the degradation under denser top-k graphs together indicate that which edges are retained, not how many, is the binding constraint." This suggests using a structural overlap threshold (J ≥ 0.30) instead of a simple edge-count threshold for structure transfer viability.

3. Enhance uncertainty quantification for dynamics-learned structures: Recognize that the condition bounds estimation error and says nothing about task-objective alignment, as Conjecture 1 (Approximation error dominance) suggests that when objectives are misaligned, approximation error dominates estimation error regardless of ∆, sample size, or threshold selection. This allows the system to predict catastrophic failure in transfer scenarios where the learned structure does not match the target task's optimal structure.

4. Implement architecture-independent structural estimators: Integrate two further dynamics-derived structure estimators (correlation- and mutual-information-based) into the localiser pipeline, as they were shown to be statistically indistinguishable from [the unstructured MLP] in terms of measurable benefit, confirming that dynamics-derived structure here provides no measurable benefit even in the one case where it does not actively hurt.

A natural strategy for inverse problems with scarce labelled data is to transfer relational structure learned from abundant forward-simulation data. We show this strategy fails systematically, even when it satisfies the standard theoretical justification for why structure should help. We prove that approximate structure provides estimation-error benefits whenever the edge error satisfies Δ< n squared - kn, reducing sample complexity from O(n 2) to O(kn+Δ). Structure learned via Neural Relational Inference (NRI) from dynamics prediction satisfies this condition, yet on a source-localisation task across 180 CFD-simulated hydrogen-leak scenarios and 180 acoustic scenarios, it degrades performance by 116% and 201% relative to a flexible, task-optimised attention baseline, while a physics-based prior (Green's function) degrades by only 69-72%. Four independent lines of evidence show this is not a tuning failure: NRI improves only 0.5% when given 18x more training data (versus 16.6% for the task-optimised baseline, p<0.001); performance is insensitive to the NRI edge threshold across a wide range; the dynamics-learned graph overlaps the task-optimal graph on only 6% of edges; and two further dynamics-derived structure estimators (correlation- and mutual-information-based) show no measurable benefit over a structure-free baseline, with the correlation-based estimator performing markedly worse. We formalise this gap as a statement about approximation error that the edge-accuracy condition cannot control, and we provide a lightweight transferability test (Jaccard similarity against a partially-observed target-task graph) that separates successful from failed transfer in all four domain/structure pairs we evaluate, using under an hour of computation and 15-20% of target-domain data; we present this as a heuristic calibrated on few cases, not a validated general threshold.