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REVIEW 3 major objections 6 minor 45 references

Soft-decay neurons by target evidence, fix PINN transfer parameters

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2026-07-07 20:17 UTC pith:CCDNI7TG

load-bearing objection Neuron-level representation correction for PINN inverse transfer learning — method is reasonable, evidence is uneven the 3 major comments →

arxiv 2607.05271 v1 pith:CCDNI7TG submitted 2026-07-06 cs.LG

Target-Guided Selective Reweighting for Physics-Informed Neural Network Inverse Problems: A Transfer Learning Approach

Qian Hu , Bin Fan , Yao Xiao , Zhicheng Lin , Meixin Xiong This is my paper
classification cs.LG
keywords targettransferselectivetgsr-pinninverselearningparametersphysical
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

When a physics-informed neural network trained on one PDE is reused for a different but related inverse problem, the transferred weights can look successful — low field-prediction error — while quietly pulling the recovered physical parameters away from their true values. The network compensates for wrong parameters by adjusting its field predictions, so the error metrics decouple: the field looks fine, the parameters are wrong. This paper proposes a method called TGSR-PINN that, after transferring only the network weights (not the source task's physical parameters), runs a short adaptation phase on the target task, scores each hidden neuron for how strongly the target evidence supports keeping it, and then gently scales down — rather than pruning or resetting — the input weights and biases of low-scoring neurons. The central mechanism is a chain: neuron target score (from Taylor sensitivity and pre-activation variance) → weak-adaptation signal (via Gaussian mixture model with rank fallback) → selective soft decay (a bounded, continuous scaling that preserves network topology). The paper claims this target-evidence-driven correction improves physical parameter recovery over full fine-tuning and layer-freezing baselines, with the benefit attributable to informed neuron selection rather than generic weight perturbation.

Core claim

The paper's central claim is that neuron-level, target-evidence-driven selective soft decay of transferred representations can reduce physical parameter recovery error in PINN inverse transfer learning, and that this benefit is not reducible to general weight perturbation. The key evidence: in a high-Péclet 2D advection-diffusion task, average parameter error drops from 6.582% (Full Fine-Tuning) to 4.426%, with best-repetition diffusion-coefficient error of 0.38% versus 4.06% for the best baseline. In Allen-Cahn-to-Burgers cross-PDE-family transfer, viscosity parameter error drops from 0.568% to 0.408%. Counterfactual ablations with 10 paired repetitions show that randomly decaying neurons (

What carries the argument

Neuron target score: a per-neuron measure combining first-order Taylor sensitivity of the target loss to the neuron's pre-activation with pre-activation variance, fused geometrically and averaged over fixed scoring batches. Weak-adaptation signal: a continuous value derived from the neuron target score distribution within each layer, estimated via a two-component Gaussian mixture model (with Bhattacharyya-based confidence and BIC-based activation) or a rank-based fallback when GMM evidence is insufficient. Selective soft decay: bounded multiplicative scaling (minimum factor 0.4) applied to the input weight row and bias of low-scoring neurons, preserving network topology and trainability. The

Load-bearing premise

The method assumes that a short 30-step target-adaptation phase produces neuron scores that reliably distinguish target-relevant from target-irrelevant neurons. If the initial physical parameters are far from their true values, this short phase may already enter a parameter compensation region, contaminating all downstream scoring, signal estimation, and decay decisions.

What would settle it

If the short-adaptation scoring state is already in a parameter-compensation region, all downstream neuron scores, weak-adaptation signals, and selective soft decay factors operate on contaminated evidence, potentially degrading rather than improving parameter recovery.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If selective soft decay works as described, PINN transfer learning for inverse problems can be made safer: practitioners could reuse source-task representations without risking silent parameter compensation, expanding the range of tasks where transfer is beneficial.
  • The decoupling of field error from parameter error suggests that the PINN community should adopt parameter-level metrics as primary evaluation criteria for inverse transfer, not just field reconstruction accuracy.
  • The target-scoring mechanism could be applied beyond transfer learning — for instance, diagnosing which neurons in a freshly trained PINN are contributing to parameter compensation versus field fitting, potentially as a training-time diagnostic tool.
  • If the short-adaptation scoring phase generalizes, the approach could extend to multi-source transfer or neural operator fine-tuning, where identifying which transferred features are target-relevant is even more critical.

Where Pith is reading between the lines

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

  • The method's reliance on a 30-step adaptation phase producing reliable neuron scores is a fragility point: if the initial physical parameter guess is far from true, the scoring signal could be contaminated, and the paper's own sensitivity analysis (Table 11) shows substantial variability with only 2 repetitions per setting — the method may benefit from an adaptive or iterative scoring approach tha
  • The rank-fallback path being the primary active route in the high-Péclet experiments (GMM rarely activates) suggests the GMM branch may be solving a problem that doesn't arise at the network widths tested (100 neurons per layer); it could become more relevant for wider networks or more heterogeneous layer distributions.
  • The piecewise cubic mapping's superiority over linear, sigmoid, and hard-threshold alternatives (Table 15) hints that the specific shape of the decay function matters less than its bounded, continuous, and monotone character — suggesting a design principle rather than a single optimal function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper proposes TGSR-PINN, a neuron-level representation correction method for PINN inverse transfer learning. After transferring only network weights/biases from a source model (not physical parameters), the method performs a short target-adaptation phase, computes neuron target scores using first-order Taylor sensitivity and pre-activation variance, estimates weak-adaptation signals via a GMM with rank fallback, and applies selective soft decay to low-scoring neurons under layer protection. The method is evaluated on three tasks: high-Péclet 2D advection-diffusion, Allen-Cahn→Burgers cross-PDE transfer, and 5%-noise reaction-diffusion. Ablation studies examine scoring components, GMM/rank fallback, layer protection, and decay mapping forms. The central claim is that target-evidence-driven neuron scoring and selective soft decay improve physical parameter recovery compared to standard fine-tuning baselines, while maintaining comparable field accuracy.

Significance. The paper addresses a genuine problem: in PINN inverse transfer learning, field accuracy and parameter recovery can decouple, and coarse-grained transfer strategies may not address neuron-level representation mismatch. The methodological pipeline is well-motivated and internally consistent. Strengths include: (1) counterfactual ablations (Table 8) with Wilcoxon tests (p≤0.037) demonstrating that the benefit is not from arbitrary perturbation; (2) layer protection ablation (Table 14, p=0.002) and decay mapping ablation (Table 15, p≤0.004) with 10 paired repetitions; (3) public code availability at a GitHub repository; (4) honest discussion of applicability boundaries, including acknowledgment that the GMM branch is not the primary active route in the high-Péclet task (§3.5.4). The paper is transparent about the large standard deviations in the high-Péclet task and positions the 5%-noise reaction-diffusion results as supplementary rather than primary evidence.

major comments (3)
  1. §3.1 and Tables 3, 5: The headline claim that TGSR-PINN improves parameter recovery over baselines rests on mean±std comparisons without pairwise statistical tests. In Table 3 (n=7), TGSR-PINN's average parameter error of 4.426±3.019% vs. Lightweight FT's 4.825±2.384% is a 0.4 percentage-point difference well within one standard deviation. In Table 5 (n=11), the ν error difference between TGSR-PINN (0.408±0.260%) and Full FT (0.568±0.399%) also has overlapping standard deviations. The paper states it uses a paired comparison protocol (§3.1), so paired data likely exists. The counterfactual ablations (Table 8) do report Wilcoxon tests, but these compare TGSR-PINN against degraded versions of itself, not against the actual baselines. Without pairwise tests against baselines (especially Lightweight FT in the high-Péclet task and Full FT in the cross-PDE task), the improvement-over-baselines
  2. Table 4 and §3.2: The prominently featured best-α error of 0.38% for TGSR-PINN is selected from 7 repetitions, while the mean α error is approximately 13.25% (per §3.5.3). The paper does note that 'core conclusions still rely on aggregated statistics,' but Table 4 is presented as a standalone comparison table and the 0.38% figure appears in the abstract-level discussion. Given the large gap between best (0.38%) and mean (~13%) α error, the best-repetition result should be more explicitly framed as a ceiling indicator rather than a representative outcome. The risk is that readers interpret 0.38% as typical performance. This is a presentation issue that affects how the central claim is perceived, but it does not invalidate the aggregated evidence.
  3. §2.2, Eqs. (11)–(21): The method has a substantial number of free hyperparameters: α_s, Δα, β_h, γ_m, γ_min, K, E_ad, ξ, ρ_r, δ_g, and the ΔBIC threshold. The paper provides sensitivity analyses for α_s (Table 12), E_ad (Table 11), and β_h (Table 14), which is commendable. However, Table 11 uses only 2 repetitions per setting, providing weak statistical support for the E_ad=30 default. The paper acknowledges this ('finer task-adaptive step selection is left for future work'), but given that E_ad controls the scoring state on which all downstream steps depend, a stronger sensitivity analysis (more repetitions, or at minimum reporting both individual repetition values rather than just two) would strengthen confidence in the default. This is not necessarily blocking since the main experiments use the default consistently across all methods, but it affects reproducibility confidence.
minor comments (6)
  1. Table 5: BitFit's field error of 69.010±13.080% and ν error of 4891±13133% are reported, but these extreme values distort any visual comparison in Figure 5 (log scale). The paper explains this as 'large-scale training instability,' but it may be worth noting whether any BitFit repetitions converged normally or whether all 11 failed, to help readers calibrate the comparison.
  2. §3.5.4, Table 13: The GMM branch is found to be inactive for most layers in the high-Péclet task (ΔBIC condition not met), making rank fallback the primary route. This is honestly reported, but it raises the question of whether the GMM component adds enough value to justify its complexity. A brief note on whether any task in the paper activates the GMM branch would help readers assess the method's complexity.
  3. Equation numbering: Eq. (19) defines A_ℓ,j and B_ℓ,j as sub-expressions, but the piecewise definition could be clearer about the transition at ep_ℓ,j = 0.5 (both branches give γ_m at that point, which is consistent but not stated).
  4. §4.3, Table 16: The computational overhead breakdown reports memory for neuron scoring as ~470 MB, which is the same as the total. This appears to be the peak memory during scoring (forward+backward on frozen model), but the table format makes it look like scoring alone requires 470 MB. A brief clarification would help.
  5. Figure 8: The aggregated scatter plot of field error vs. parameter error is informative, but the error bars for TGSR-PINN and Lightweight FT overlap substantially in both dimensions. A note acknowledging this overlap would make the figure's interpretation more consistent with the text's cautious framing.
  6. The paper would benefit from a brief comparison to existing neuron-level pruning or importance estimation methods in the PINN context (e.g., Ref. [35] is cited but the comparison is brief). A paragraph clarifying what specifically differs from selective pruning for PINN inverse problems would strengthen the novelty claim.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The three major comments are well-taken: (1) pairwise statistical tests against baselines are absent in Tables 3 and 5 and should be added; (2) the best-repetition alpha error of 0.38% in Table 4 needs clearer framing relative to the mean of ~13%; and (3) the E_ad sensitivity analysis in Table 11 uses only 2 repetitions, which is weak. We agree with all three points and will revise accordingly. The referee's positive assessment of the counterfactual ablations, layer-protection and decay-mapping experiments, code availability, and honest discussion of applicability boundaries is appreciated.

read point-by-point responses

  1. Referee: §3.1 and Tables 3, 5: The headline claim that TGSR-PINN improves parameter recovery over baselines rests on mean±std comparisons without pairwise statistical tests. In Table 3 (n=7), TGSR-PINN's average parameter error of 4.426±3.019% vs. Lightweight FT's 4.825±2.384% is a 0.4 percentage-point difference well within one standard deviation. In Table 5 (n=11), the ν error difference between TGSR-PINN (0.408±0.260%) and Full FT (0.568±0.399%) also has overlapping standard deviations. The paper states it uses a paired comparison protocol (§3.1), so paired data likely exists. The counterfactual ablations (Table 8) do report Wilcoxon tests, but these compare TGSR-PINN against degraded versions of itself, not against the actual baselines. Without pairwise tests against baselines (especially Lightweight FT in the high-Péclet task and Full FT in the cross-PDE task), the improvement-over-baselines

    Authors: The referee is correct. We have paired data from the matched-repetition protocol described in §3.1, and we should have reported pairwise statistical tests against the actual baselines in Tables 3 and 5, not only in the counterfactual ablations (Table 8). We will add Wilcoxon signed-rank tests for the key baseline comparisons: TGSR-PINN vs. Lightweight FT in the high-Péclet task (n=7 paired) and TGSR-PINN vs. Full FT in the cross-PDE task (n=11 paired), as well as the other baseline comparisons. We will report p-values and effect sizes alongside the existing mean±std entries. We acknowledge in advance that some comparisons—particularly the 0.4 percentage-point gap between TGSR-PINN and Lightweight FT in the high-Péclet task—may not reach statistical significance at conventional thresholds given n=7 and the large standard deviations. If that is the case, we will state it plainly and qualify the corresponding claim rather than overstate it. The cross-PDE comparison (n=11, 28.2% relative reduction in ν error) is more likely to reach significance, but we will let the test results speak for themselves. We will also add a note clarifying that the counterfactual ablation tests (Table 8) were designed to answer a different question—whether the benefit comes from target-informed neuron selection versus arbitrary perturbation—rather than to substitute for baseline comparisons. revision: yes

  2. Referee: Table 4 and §3.2: The prominently featured best-α error of 0.38% for TGSR-PINN is selected from 7 repetitions, while the mean α error is approximately 13.25% (per §3.5.3). The paper does note that 'core conclusions still rely on aggregated statistics,' but Table 4 is presented as a standalone comparison table and the 0.38% figure appears in the abstract-level discussion. Given the large gap between best (0.38%) and mean (~13%) α error, the best-repetition result should be more explicitly framed as a ceiling indicator rather than a representative outcome. The risk is that readers interpret 0.38% as typical performance. This is a presentation issue that affects how the central claim is perceived, but it does not invalidate the aggregated evidence.

    Authors: We agree this is a presentation issue that needs correction. The 0.38% best-repetition α error is a ceiling indicator showing what target-side representation correction can achieve under favorable optimization conditions, not a representative outcome. The mean α error of ~13.25% (reported in §3.5.3) is the representative statistic. We will make the following changes: (1) relabel Table 4 to explicitly state 'best-repetition (ceiling indicator)' in the caption; (2) add the mean α error alongside the best α error in Table 4 so readers see both figures; (3) revise the §3.2 discussion to state upfront that the mean α error is approximately 13.25% and that 0.38% represents the best repetition, not typical performance; and (4) ensure that any abstract-level or summary discussion references the mean rather than the best. We appreciate the referee noting that this does not invalidate the aggregated evidence—the core claims in Tables 3 and 5 are based on mean±std across repetitions, and the counterfactual ablations in Table 8 use 10 paired repetitions with Wilcoxon tests. revision: yes

  3. Referee: §2.2, Eqs. (11)–(21): The method has a substantial number of free hyperparameters: α_s, Δα, β_h, γ_m, γ_min, K, E_ad, ξ, ρ_r, δ_g, and the ΔBIC threshold. The paper provides sensitivity analyses for α_s (Table 12), E_ad (Table 11), and β_h (Table 14), which is commendable. However, Table 11 uses only 2 repetitions per setting, providing weak statistical support for the E_ad=30 default. The paper acknowledges this ('finer task-adaptive step selection is left for future work'), but given that E_ad controls the scoring state on which all downstream steps depend, a stronger sensitivity analysis (more repetitions, or at minimum reporting both individual repetition values rather than just two) would strengthen confidence in the default. This is not necessarily blocking since the main experiments use the default consistently across all methods, but it affects reproducibility confidence.

    Authors: The referee's point is valid. E_ad controls the post-adaptation state on which all downstream scoring and selective soft decay depend, so Table 11's 2-repetition sensitivity analysis is insufficient. We will expand this to at least 5 repetitions per E_ad setting (E_ad ∈ {10, 20, 30, 60}) and report individual repetition values alongside the mean. We already report both individual values (Rep A, Rep B) in the current table, so the expanded version will follow the same format with more repetitions. We note that the main experiments use E_ad=30 consistently across all methods, so this does not affect the validity of the main comparisons; the issue is specifically about confidence in the default choice. We also agree that the large number of hyperparameters is a legitimate concern. While we provide sensitivity analyses for the most impactful ones (α_s in Table 12 with 5 reps, β_h in Table 14 with 10 reps, decay mapping in Table 15 with 10 reps), we will add a consolidated hyperparameter table in the revised manuscript listing all defaults, their roles, and which (if any) sensitivity analysis covers them, so readers can assess the overall robustness picture at a glance. revision: yes

Circularity Check

0 steps flagged

No circularity found: the derivation chain transforms target-task evidence into neuron scores, decay signals, and weight scaling, with final results evaluated against externally known true parameters.

full rationale

The paper's derivation chain proceeds as: (1) target loss gradients and pre-activation statistics → Taylor sensitivity (Eq. 9) and pre-activation variance (Eq. 10); (2) geometric fusion → neuron target score (Eq. 11); (3) GMM/rank fallback → weak-adaptation signal (Eqs. 16–17); (4) layer protection + piecewise cubic mapping → scaling factor γ (Eqs. 18–19); (5) weight scaling (Eq. 21) → continued training → parameter recovery evaluated against known true values. At no point is a quantity defined in terms of the result it claims to predict. The neuron target score is computed from target loss gradients and pre-activation responses—quantities derived from the target task, not from the parameter recovery outcome. The weak-adaptation signal is derived from the score distribution, not from the final parameter error. The selective soft decay factors are derived from the weak-adaptation signal, not from the claimed improvement. All citations supporting methodological choices (Taylor sensitivity [32], GMM/EM [40], Bhattacharyya coefficient [41], BIC [42], soft pruning [43–45]) are to external authors, not self-citations. The experimental evaluation uses externally known true physical parameters (α, ν, etc.) as ground truth, and baselines (Full FT, Lightweight FT, BitFit, etc.) are independent methods. Hyperparameters (α_s, β_h, γ_m, γ_min) are selected via ablation sweeps (Tables 12, 14, 15), which is standard model selection, not circularity. No step in the derivation reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

12 free parameters · 5 axioms · 3 invented entities

The paper introduces 12 free parameters, most chosen empirically without independent ablation (γ_m, γ_min, ξ, ρ_r, Δα are not individually ablated). The key parameters that ARE ablated (α_s, β_h, mapping form) show statistically significant effects. The 5 axioms are domain assumptions about the adequacy of first-order Taylor expansion, pre-activation statistics, short-adaptation reliability, GMM modeling, and decay placement. The 3 invented entities (neuron target score, weak-adaptation signal, selective soft decay) all have independent falsifiable handles through counterfactual ablations. The circularity burden is low because the method is evaluated against external benchmarks (known true parameters) and compared to independent baselines.

free parameters (12)
  • α_s (base Taylor/variance fusion weight) = 0.5
    Controls relative weight of Taylor sensitivity vs. pre-activation variance in neuron scoring (Eq. 11). Swept over [0,1] in Table 12; 0.5 chosen as default within low-error plateau [0.3, 0.8].
  • Δα (layer-wise fusion increment) = 0.3
    Controls how fusion weight shifts across layers (Eq. 11). Deeper layers score more toward Taylor sensitivity. Not independently ablated; chosen by design.
  • β_h (shallow protection strength) = 0.55
    Controls layer protection coefficient (Eq. 18). Ablated in Table 14 (β_h=0 vs. 0.55, p=0.002). Value chosen empirically.
  • γ_m (intermediate scaling factor) = 0.85
    Scaling factor boundary in piecewise cubic mapping (Eq. 19). Not independently ablated; chosen by design.
  • γ_min (minimum scaling factor) = 0.4
    Lower bound on neuron scaling (Eq. 19). Not independently ablated; chosen by design.
  • K (number of scoring batches) = 3
    Number of fixed scoring batches (Eq. 8). Ablated in Table 9 (K=1 vs. 3, minimal difference). Default chosen empirically.
  • E_ad (target short adaptation steps) = 30
    Length of short adaptation phase (Eq. 7). Set as 20% of main training. Sensitivity analyzed in Table 11 with 2 repetitions.
  • ξ (rank fallback max signal) = 0.75
    Maximum weak-adaptation signal in rank fallback (Eq. 17). Not ablated; chosen by design.
  • ρ_r (rank power exponent) = 1.5
    Controls nonlinearity of rank-based signal (Eq. 17). Not ablated; chosen by design.
  • δ_g (reporting threshold) = 0.02
    Threshold for diagnostic set of effectively softened neurons (Eq. 20). Only for reporting, not operational.
  • ΔBIC threshold = 10
    BIC difference threshold for GMM branch activation. Standard model selection criterion.
  • π_{ℓ,L} threshold = 0.5
    Maximum mixture weight for low-mean component to activate GMM branch. Ensures low-scoring neurons are a minority.
axioms (5)
  • domain assumption First-order Taylor expansion adequately approximates neuron importance for the target loss
    Eq. (9) uses first-order Taylor sensitivity z·∂L/∂z as neuron importance. This is standard in pruning literature [32] but ignores second-order and higher terms. Invoked in Section 2.2 when defining Taylor sensitivity.
  • domain assumption Pre-activation variance characterizes neuron response activity on target samples
    Eq. (10) uses pre-activation (not post-activation) variance as auxiliary evidence. The paper justifies this by referencing initialization/normalization literature [37-39] on signal propagation, but these works address training stability, not neuron importance for inverse problems.
  • ad hoc to paper A short adaptation phase produces reliable target evidence for neuron scoring
    The entire pipeline depends on Θ^ad_T being a meaningful diagnostic state. Section 2.2 acknowledges this is fragile: 'If this phase is too long or the initial physical parameters deviate severely from true values, the scoring signal may also be contaminated.' Table 11 provides only 2-repetition sensitivity analysis.
  • domain assumption Two-component GMM adequately models intra-layer neuron target score distributions
    Eq. (14) assumes scores decompose into low-mean and high-mean components. In practice (Table 13), most layers do not meet the ΔBIC>10 condition, so this assumption is frequently violated and rank fallback is used instead.
  • domain assumption Soft decay of pre-activations is preferable to post-activation gating or outgoing weight scaling
    Section 2.2 (after Eq. 21) states this choice is made because it 'requires no additional network modules or permanent masks and matches the parameterization used in the implementation.' The paper notes non-equivalence under tanh but does not compare alternatives empirically.
invented entities (3)
  • Neuron target score independent evidence
    purpose: Composite metric (Taylor sensitivity × pre-activation variance) measuring neuron relevance to target task
    The score is computed from target loss gradients and forward-pass statistics on fixed scoring batches. Counterfactual ablation (Table 8: Score-Shuffled, p=0.014) shows that preserving score-neuron correspondence matters, providing falsifiable evidence that the score carries target-specific information.
  • Weak-adaptation signal independent evidence
    purpose: Continuous signal derived from neuron target scores indicating how much to decay each neuron
    The signal is derived from the score distribution via GMM or rank fallback. The mapping form ablation (Table 15) shows the piecewise cubic mapping outperforms alternatives (p≤0.004), providing evidence that the signal-to-decay mapping matters.
  • Selective soft decay independent evidence
    purpose: Continuous scaling of input weight rows and biases of low-scoring neurons
    The operation (Eq. 21) is distinguished from hard pruning and random resetting. Table 7 shows Random Reset (param error 19.457%) vs. TGSR-PINN (2.555%), and Table 8 shows Random Soft Decay is significantly worse (p=0.037).

pith-pipeline@v1.1.0-glm · 27543 in / 4697 out tokens · 185775 ms · 2026-07-07T20:17:11.558424+00:00 · methodology

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read the original abstract

Physics-informed neural networks (PINNs) encounter ill-posed optimization, loss competition, and parameter compensation in partial differential equation (PDE) inverse problems. Transfer learning can reuse representations from source tasks, but direct fine-tuning may introduce negative transfer when dominant physical mechanisms, governing parameters, or observation noise differ between source and target domains: the model achieves low field error yet recovers incorrect target physical parameters. To mitigate, we propose Target-Guided Selective Reweighting PINN (TGSR-PINN), a target-evidence-driven representation correction method for PINN inverse transfer learning. TGSR-PINN transfers only the weights and biases from the source PINN, while target physical parameters are independently initialized; after a short target-adaptation phase, the method computes neuron target scores using first-order Taylor sensitivity and pre-activation variance on fixed scoring batches, and converts evidence associated with low-scoring neurons into continuous weak-adaptation signals via a Gaussian mixture model (GMM) with rank fallback. TGSR-PINN then applies selective soft decay to input weight rows and biases of low-scoring neurons instead of hard pruning or random resetting. In experiments, TGSR-PINN improves target parameter recovery while maintaining comparable field accuracy in the high-P\'{e}clet 2D advection-diffusion task and in the Allen--Cahn to Burgers cross-PDE-family transfer task; a 5%-noise reaction--diffusion case provides supplementary evidence under milder source-target mismatch. Ablation studies suggest that neuron target scoring, weak-adaptation signal estimation, layer protection, and selective soft decay jointly contribute to the benefits.

Figures

Figures reproduced from arXiv: 2607.05271 by Bin Fan, Meixin Xiong, Qian Hu, Yao Xiao, Zhicheng Lin.

Figure 1
Figure 1. Figure 1: TGSR-PINN method pipeline. 2.1. PINN Inverse Problem Formulation We first present the general PINN formulation for PDE in￾verse problems. Consider a physical system defined on a spatial domain Ω and a temporal interval [0, T]. We directly use spa￾tial coordinates x and time t as network inputs; for steady-state problems, the time variable t can be omitted. Let u(x, t) denote the physical field to be solved… view at source ↗
Figure 2
Figure 2. Figure 2: High-Péclet task training loss curves [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: High-Péclet task parameter error convergence curves [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: High-Péclet task t = 1 cross-section: true field and absolute error maps [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Repetition-level distribution of ν relative errors in Allen–Cahn → Burgers cross-PDE transfer [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: αs parameter sweep curve. The αs ∈ [0.3, 0.8] range forms a low￾error plateau, with higher errors at both ends. 3.5.4. GMM and rank fallback ablation TGSR-PINN employs a dual-path design with GMM distri￾bution diagnosis and rank fallback for weak-adaptation signal estimation. The core idea is: when the intra-layer neuron tar￾get score distribution exhibits a clear separation between low￾scoring and high-sc… view at source ↗
Figure 8
Figure 8. Figure 8: Aggregated scatter plot of field error vs. parameter error on the high [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Parameter error convergence curves on the high-Péclet task [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Parameter error changes before and after target short adaptation and [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Layer-wise selective soft decay diagnosis for layer protection and [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗

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