A Graph-Based Inspection and Intervention Tool for Assessing Mechanistic Learning in PINNs

Adwait Patkhedkar; Alifaraz Lakhani; Prathmesh Mohite; Abhijeet Salunke

cs.LGOct 4, 2026

A Graph-Based Inspection and Intervention Tool for Assessing Mechanistic Learning in PINNs

Authors: Adwait Patkhedkar, Alifaraz Lakhani, Prathmesh Mohite, Abhijeet Salunke

Organizations: Sardar Patel Institute of Technology, Mumbai, India

Abstract

We ask whether physically meaningful correspondences discovered inside a trained scientific model remain meaningful outside the conditions under which they were discovered. We introduce GIIT (Graph-based Inspection and Intervention Tool), which represents governing physics as a computational physics dependency graph, maps graph nodes to internal network components via sensitivity- and trend-based discovery, and tests the resulting mapping under targeted intervention. Evaluating on temporal extrapolation out-of-distribution (OOD) regimes of increasing severity, we find that temporal extrapolation is associated with layer-wise correspondence drift and reduced functional correspondence. Specifically, while the model achieves low physics residual in-distribution (1.915 x 10-4 on full ID and 1.01 x 10-4 on an ID sub-window t in [0.5, 0.8]), the functional mapping changes even before leaving the training domain, and the shift continues as the evaluation window extends beyond the training domain: residual error increases from 2.81 x 10-2 on the boundary-crossing window (t in [0.5, 1.5]) to 3.30 x 10-1 on severe OOD (t in [1, 2]), accompanied by a systematic leftward shift of internal layer mappings, where the average winner layer index drops from 5.71 (ID) to 3.00 (ID sub-window) down to 2.29 (severe OOD). Only 2 of 7 physical nodes maintain stable layer assignments across the boundary-crossing window, and only 1 of 7 under severe extrapolation, revealing potential internal functional correspondence changes before severe degradation in conventional output metrics. Additional results for linear oscillatory systems are further detailed in the appendix.

Figures & tables

Figure 1: GIIT pipeline: Physical specification $\rightarrow$ Physics graph $\rightarrow$ Neural model $\rightarrow$ Variable/parameter mapping $\rightarrow$ Intervention $\rightarrow$ Physical observables $\rightarrow$ ID $\to$ OOD transfer test.

Regime	$L_{2}$ (%)	Residual $\text{mean}(R^{2})$	$\partial\mathcal{L}_{R}/\partial\nu$	Stable Mappings
ID ( $t\in[0,1]$ )	$3.855$	$1.915\times 10^{-4}$	$-1.705$	— (baseline)
ID Sub-window ( $t\in[0.5,0.8]$ )	$5.413$	$1.011\times 10^{-4}$	$-5.064$	2/7 (28.6%)
Boundary-crossing ( $t\in[0.5,1.5]$ )	$5.883$	$2.811\times 10^{-2}$	$-73.576$	2/7 (28.6%)
OOD ( $t\in[1,2]$ )	$31.752$	$3.301\times 10^{-1}$	$-154.115$	1/7 (14.3%)

Table 1: Viscous Burgers’ equation: ID (full and sub-window), boundary-crossing, and OOD extrapolation windows (

t\in[1,2]

). A node mapping is defined as stable if its modal winning layer matches the full-ID baseline.

Physical Node	ID ( $t\in[0,1]$ )		ID Sub ( $[0.5,0.8]$ )		Bound.-cross ( $[0.5,1.5]$ )		OOD ( $[1,2]$ )
	Top	Cons.	Top	Cons.	Top	Cons.	Top	Cons.
Kinematic Viscosity ( $\nu$ )	h5w	69/100	h5w	67/100	h5w	56/100	h0w	73/100
Velocity Field ( $u$ )	h5w	35/100	h5w	33/100	h5w	30/100	h5w	29/100
Temporal Gradient ( $u_{t}$ )	h6w	34/100	h5w	23/100	h1w	24/100	h5w	25/100
Spatial Gradient ( $u_{x}$ )	h6w	23/100	h1w	22/100	h1w	24/100	h1w	30/100
Spatial Laplacian ( $u_{xx}$ )	h6w	31/100	h0w	26/100	h0w	35/100	h0w	35/100

Table 2: Neural Component Mapping Distributions across 100 Perturbation Replicates on One Trained Model (

\sigma=0.1

, replicates

\in[0,99]

) for ID (full and sub-window) and extrapolation regimes. The average winner layer index (bottom row) is the mean modal hidden layer index (

k\in[0,6]

for h

k

w ) across all 7 physical nodes. Abbreviations: h

k

=

fch.

k

.0.weight .

Panel of Figure 2 — Figure 2: GIIT layer mapping distributions across 100 perturbation replicates ( $\sigma=0.1$ ) for Burgers’ equation under ID and OOD regimes. Cell values show winner counts per layer, highlighting systematic leftward migration ( $5.71\to 2.29$ ).

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: 3D comparison for the 1D viscous Burgers’ velocity field across the in-distribution space-time domain $(t,x)\in[0,1]\times[-1,1]$ : (a) high-resolution finite-difference reference solution $u(t,x)$ , (b) PINN base model prediction $\hat{u}(t,x)$ ( $L_{2}\approx 3.86\%$ ), and (c) pointwise absolute error $|\hat{u}(t,x)-u(t,x)|$ , illustrating that predictive errors are primarily concentrated along the steep internal shock layer at $x\approx 0$ .

Figure 4: 3D comparison for the 1D viscous Burgers’ velocity field under out-of-distribution temporal extrapolation ( $t\in[1,2],x\in[-1,1]$ ): (a) high-resolution reference solution $u(t,x)$ continuing smooth viscous dissipation, (b) PINN model prediction $\hat{u}(t,x)$ showing severe unphysical deformation ( $L_{2}\approx 31.75\%$ ), and (c) pointwise absolute error $|\hat{u}(t,x)-u(t,x)|$ , highlighting large localized error growth along the extrapolated shock front.

Regime	Pred. $L_{2}$ error	Physics residual	Layer correspondence
ID ( $t\in[0,1]$ )	$1.13\%$	$2.007\times 10^{2}$	Distinct parameter mappings identified
OOD ( $t\in[1,2]$ )	$2007.34\%$	$1.317\times 10^{5}$	Mass/damping collapse to shared tensor; spring shifts

Appendix

Table 3: Oscillator results across In-Distribution (

t\in[0,1]

) and Out-Of-Distribution (

t\in[1,2]

) regimes.

Figure 5: Damped harmonic oscillator dynamics in-distribution ( $t\in[0,1]$ ) and under extrapolation ( $t\in[1,2]$ ): (a) displacement $x(t)$ comparing ground truth against the PINN, which diverges to a flatline ( $\hat{x}\approx-1.19$ ) after $t=1.0$ ; (b) phase portrait ( $\dot{x}$ vs. $x$ ) showing the true trajectory spiraling into the origin $(0,0)$ , whereas the PINN terminates at $(-1.19,-0.19)$ .

Physical Parameter	ID ( $t\in[0,1]$ )	OOD ( $t\in[1,2]$ )
Mass ( $m$ )	$+1377.28$	$-1747.92$
Damping Coeff ( $c$ )	$-3.84$	$+579.62$
Spring Constant ( $k$ )	$+111.77$	$+458.12$

Appendix

Table 4: Residual-loss Sensitivities (

\partial\mathcal{L}_{R}/\partial p

Physical Variable	In-Distribution ( $t\in[0,1]$ )		Out-Of-Distribution ( $t\in[1,2]$ )
	Top Component	Consistency	Top Component	Consistency
Mass ( $m$ )	fch.1.0.weight	10/21 (47.6%)	fch.0.0.weight	16/21 (76.2%)
Damping Coeff ( $c$ )	fch.1.0.weight	10/21 (47.6%)	fch.0.0.weight	16/21 (76.2%)
Spring Constant ( $k$ )	fch.0.0.weight	8/21 (38.1%)	fch.1.0.weight	12/21 (57.1%)
Displacement ( $x$ )	fch.1.0.weight	11/21 (52.4%)	fch.1.0.weight	16/21 (76.2%)
Velocity ( $\dot{x}$ )	fch.0.0.weight	17/21 (81.0%)	fch.0.0.weight	13/21 (61.9%)

Appendix

Table 5: Neural Component Mapping Distributions across 21 Perturbation Replicates (

\text{replicate}\in[0,20]

\sigma=0.1

) for both In-Distribution (

t\in[0,1]

) and Out-Of-Distribution (

t\in[1,2]

) evaluation.

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LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks

Nilay Anurag, Shital Adhikari, Taniya Kapoor, Nikhil Muralidhar

Department of Computer Science, Stevens Institute of Technology · Artificial Intelligence Group, Wageningen University & Research

Neural PDE Solvers Neural Network Optimization Physics-Informed ML

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Physics-informed neural networks (PINNs) have had a broad research impact in modeling domains governed by partial differential equations (PDE). However, PINNs have been shown to perform poorly, sometimes even converging to trivial solutions, in challenging PDE domains, or when generalizing to unseen but related PDE domains. Previously proposed solutions detail hyperparameter tuning to reduce loss imbalance between data-driven and physics guided losses, curriculum learning based training strategies, or dynamic re-sampling of hard collocation points. These methods face certain pitfalls: hyperparameter tuning is expensive, designing a training curriculum is ambiguous in multi-parameter PDE settings, and dynamic resampling still fails in complex PDE settings. Complementary to this line of thinking, we believe the initial PINN network weights also play a crucial role in the emergence of catastrophic failures during training, yet the effect of PINN weight initialization has been surprisingly under-investigated. To this end, we propose a framework for Learned Initialization via Gated Layerwise Optimization (LIGO-PINN) to overcome PINN convergence failures. Through rigorous evaluation on 1D and 2D PDE domains, including a challenging 2D fluid dynamics setting, we demonstrate that our methodology outperforms state-of-the-art methods designed to alleviate PINN failures, achieving a 91.5% average performance improvement across six baselines and 81% over the strongest baseline. We also verify that LIGO-PINN generalizes to 3D unstructured domains. Finally, we analyze training dynamics across all three PDE domains to explain both LIGO-PINN's improvement and the convergence failure of traditional PINNs. Code: https://github.com/scailab/ligo-pinn Keywords: Machine Learning, Physics-Informed Neural Networks, Deep Learning, PDE Modeling

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