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arxiv: 2604.06534 · v1 · submitted 2026-04-08 · 📡 eess.SP · eess.IV

FOSSA: First-Order Optimality-Based Sensor Selection for PINN Inverse Problems, with Application to Electrocardiographic Imaging

Jianxin Xie This is my paper

Pith reviewed 2026-05-10 18:40 UTC · model grok-4.3

classification 📡 eess.SP eess.IV
keywords physics-informed neural networkssensor selectioninverse problemsfirst-order optimalityelectrocardiographic imagingPINN
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The pith

Sensor importance for PINN inverse problems is revealed by first-order optimality after one training run.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes FOSSA to assign importance scores to candidate sensor locations for physics-informed neural network inverse problems. It does so by evaluating the first-order optimality condition once the network has converged, avoiding the need for multiple training iterations. This global assessment shows which sensors contribute to better reconstructions and which do not. Validation on electrocardiographic imaging demonstrates that including low-importance sensors can actually reduce accuracy. A refinement scheme is added to handle cases where the inverse solver is unstable.

Core claim

FOSSA evaluates the first-order optimality condition of the converged PINN to produce importance scores for all candidate sensors in a post-training step. This enables identification of sensors that positively or negatively affect the solution quality. In the electrocardiography application, the scores indicate that some sensors degrade performance when incorporated into the model.

What carries the argument

The first-order optimality condition at convergence of the PINN training, used to compute per-sensor importance scores from the data loss term.

If this is right

  • A single converged PINN model allows ranking of every possible sensor location.
  • Sensors identified as low-importance can be omitted to avoid degrading reconstruction accuracy.
  • The method avoids the computational expense of iterative sensor addition and retraining.
  • Global sensor characterization supports better deployment decisions in physics-based inverse modeling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If sensors interact strongly, the first-order scores may underestimate combined effects and require validation through targeted experiments.
  • The refinement scheme for instability could be generalized to other optimization challenges in PINNs.
  • This post-hoc analysis opens the possibility of designing sensor networks optimized for the specific physics of the problem rather than uniform placement.

Load-bearing premise

That evaluating the first-order optimality condition at a single converged solution gives a reliable measure of each sensor's marginal contribution without considering higher-order effects or sensor interactions.

What would settle it

Training the PINN with and without the lowest-ranked sensors from FOSSA and observing whether the reconstruction error in the electrocardiographic imaging problem decreases when low-importance sensors are excluded.

Figures

Figures reproduced from arXiv: 2604.06534 by Jianxin Xie.

Figure 1
Figure 1. Figure 1: Illustration of the proposed FOSSA-integrated physics-constrained inverse ECG framework. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reproducibility of FOSSA-derived importance maps under different noise levels and training conditions. Each subfigure corresponds to a noise setting: [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstruction performance under different FOSSA-ranked observation sets and noise levels. (a) [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative comparison of heart-surface potential reconstruction at a [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of sensor selection strategies for inverse ECG reconstruction under varying sensing budgets ( [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
read the original abstract

Physics-informed neural networks (PINNs) have emerged as a powerful framework for modeling physical systems and solving inverse problems. In such settings, sensors are deployed to capture observable system responses; however, the quality of reconstruction critically depends on how these sensors are selected. Existing sensor selection strategies for PINNs are closely related to active learning and experimental design, typically relying on iterative refinement schemes that sequentially add sensors and retrain the model. While effective under limited data regimes, these approaches incur substantial computational cost due to repeated retraining and primarily focus on selecting subsets of sensors, without providing a global characterization of sensor importance. In this work, we propose FOSSA, a first-order optimality-based sensor selection algorithm for inverse PINNs. Unlike existing methods, FOSSA evaluates sensor importance in a post-training manner, requiring only a single trained PINN. FOSSA assigns importance scores to all candidate sensing locations based on the first-order optimality condition at convergence. To improve robustness, a refinement scheme is further proposed to handle instability in the inverse solver. FOSSA facilitates a global assessment of the contribution of each sensor to reconstruction. We validate the proposed approach on the inverse electrocardiography (ECG) modeling and show that not all sensors contribute positively to predictive performance. Incorporating low-importance sensors can, in fact, degrade reconstruction accuracy. These findings highlight the need for principled sensor importance evaluation and provide a scalable pathway for guiding sensor deployment in physics-informed inverse modeling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript proposes FOSSA, a post-training sensor selection method for PINN inverse problems. It computes per-sensor importance scores directly from the first-order optimality (stationarity) residual of the converged PINN loss after a single training run, avoiding the repeated retraining required by active-learning baselines. A refinement step is added to mitigate solver instability. On the inverse ECG problem the authors report that low-importance sensors can degrade reconstruction accuracy, providing a global ranking of candidate electrode locations.

Significance. If validated, the approach supplies a low-cost, global diagnostic for sensor utility in non-convex PINN inverse problems—an attractive alternative to iterative experimental-design loops. The ECG demonstration that “more sensors can hurt” is practically relevant for biomedical imaging where electrode placement is costly. The single-run character and explicit use of the optimality condition are clear methodological strengths, provided the scores are shown to predict actual marginal error changes.

major comments (3)
  1. [§3.2] §3.2 (importance-score definition): the score is extracted from the first-order stationarity condition evaluated at one converged point. Because the PINN loss is non-convex and the forward operator produces spatially correlated measurements, the marginal effect of adding or removing a sensor is generally a higher-order quantity. The manuscript must demonstrate, via leave-one-out or incremental retraining experiments, that low-scoring sensors actually increase reconstruction error and that the first-order ranking correlates with these changes.
  2. [§4] §4 (ECG experiments): the central claim that “incorporating low-importance sensors can degrade reconstruction accuracy” is load-bearing yet unsupported by the quantitative detail supplied in the abstract. The full manuscript must report concrete metrics—relative L2 error, RMSE, or correlation coefficients—with and without the low-importance subset, together with standard deviations over multiple random seeds or initializations and a direct comparison against random or greedy baselines.
  3. [§3.3] §3.3 (refinement scheme): the procedure for handling instability is described but its effect on the resulting importance scores is not quantified. An ablation showing score stability with and without refinement, and the number of additional forward/adjoint solves required, is needed to establish that the method remains practical and that the scores remain reliable.
minor comments (3)
  1. The abstract would be strengthened by including one or two key quantitative results (e.g., error reduction percentages) so that readers can immediately gauge the practical impact.
  2. [Notation] Notation for the per-sensor residual and the weighting of data-fidelity terms should be made fully explicit and cross-referenced to the PINN loss in Eq. (X).
  3. [Introduction] A short discussion of related sensor-selection literature in inverse problems (e.g., optimal experimental design for linear inverse problems) would help situate the contribution.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive review. We appreciate the recognition of FOSSA's methodological strengths as a single-run, optimality-based diagnostic and the practical relevance of the ECG findings. We address each major comment below and will incorporate the requested validations and quantitative details in the revision.

read point-by-point responses

  1. Referee: §3.2 (importance-score definition): the score is extracted from the first-order stationarity condition evaluated at one converged point. Because the PINN loss is non-convex and the forward operator produces spatially correlated measurements, the marginal effect of adding or removing a sensor is generally a higher-order quantity. The manuscript must demonstrate, via leave-one-out or incremental retraining experiments, that low-scoring sensors actually increase reconstruction error and that the first-order ranking correlates with these changes.

    Authors: We agree that non-convexity and measurement correlations imply that first-order scores are an approximation whose predictive power for marginal error changes should be verified. The current manuscript provides supporting evidence through the observed degradation in ECG reconstructions when low-importance sensors are retained, but we acknowledge that direct correlation analysis via retraining would strengthen the validation. We will add leave-one-out and incremental retraining experiments on representative sensor subsets in the revised §4 (or a new appendix) to quantify error increases for low-scoring sensors and report Pearson/Spearman correlations between FOSSA scores and the resulting error deltas. revision: yes

  2. Referee: §4 (ECG experiments): the central claim that “incorporating low-importance sensors can degrade reconstruction accuracy” is load-bearing yet unsupported by the quantitative detail supplied in the abstract. The full manuscript must report concrete metrics—relative L2 error, RMSE, or correlation coefficients—with and without the low-importance subset, together with standard deviations over multiple random seeds or initializations and a direct comparison against random or greedy baselines.

    Authors: The manuscript reports quantitative reconstruction results in §4 that demonstrate degradation when low-importance sensors are included. To meet the referee's request for fuller detail, we will revise §4 to include explicit tables/figures with relative L2 errors, RMSE, and correlation coefficients comparing the full sensor set against the high-importance subset. We will also report standard deviations over multiple random seeds/initializations and add direct comparisons against random selection and a greedy active-learning baseline to benchmark performance. revision: yes

  3. Referee: §3.3 (refinement scheme): the procedure for handling instability is described but its effect on the resulting importance scores is not quantified. An ablation showing score stability with and without refinement, and the number of additional forward/adjoint solves required, is needed to establish that the method remains practical and that the scores remain reliable.

    Authors: We agree that quantifying the refinement's effect is necessary to confirm practicality and reliability. The refinement is a lightweight post-processing step to stabilize the stationarity residual. In the revision we will add an ablation in §3.3 comparing importance-score distributions, rankings, and variance with versus without refinement. We will also tabulate the average number of additional forward and adjoint solves incurred by the refinement across the ECG experiments, showing that the overhead remains modest while improving score stability. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The FOSSA method derives sensor importance scores directly from the first-order stationarity condition of the PINN loss evaluated at a single converged solution, then empirically validates on ECG data that low-scoring sensors can degrade reconstruction accuracy when included. This is a heuristic proposal rather than a self-definitional loop: the score is computed from the trained model's optimality residual, but the accuracy degradation claim is tested via separate subset experiments that are not forced by the score definition itself. No load-bearing self-citations, ansatzes smuggled via prior work, or fitted parameters renamed as independent predictions appear in the abstract or described chain. The derivation remains self-contained as an algorithmic contribution with external empirical checks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the assumption that a converged PINN satisfies first-order optimality and that this local condition can be linearly decomposed across sensor locations; no free parameters or new physical entities are introduced in the abstract.

axioms (1)
  • domain assumption A trained PINN has reached a stationary point where the first-order optimality condition holds for the composite loss.
    Invoked to justify using the optimality residual as an importance measure.

pith-pipeline@v0.9.0 · 5564 in / 1182 out tokens · 26434 ms · 2026-05-10T18:40:54.890342+00:00 · methodology

discussion (0)

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