A two-axis post-training diagnosis is introduced that separates finite-sample resolution under a specified observation-and-estimation protocol from the signed parameter preference encoded by the final learned field and residual metric.
Abstract
Inverse physics-informed neural networks (PINNs) can reconstruct a field accurately while returning an incorrect physical parameter. We introduce a two-axis post-training diagnosis that separates finite-sample resolution under a specified observation-and-estimation protocol from the signed parameter preference encoded by the final learned field and residual metric. The first axis repeatedly fits noisy observations with a matched forward estimator. At known synthetic truth, the second freezes the field and residual view and computes a local score displacement toward a nearby residual-profile minimum. Endpoint consistency then tests whether joint training delivers that preference under the same final view. Across three synthetic one-dimensional, scalar-parameter PDEs, matched-forward mean absolute relative error ranges from 2.34 percent to 17.46 percent. The displacement tracks frozen-profile minima across locked seeds, architectures, and fresh-noise retraining (r from .945 to .982), and it tracks delivered signed log-error in 240 fresh-noise RBA runs (r = .994; 237/240 correct directions). A coupled two-parameter Darcy check validates the full matrix calculation. The axes are complementary diagnostic coordinates, not additive error components or a deployable oracle-free estimator. Together, they route follow-up work toward observations, residual evidence, or endpoint delivery.
The results demonstrate that PINN achieves more accurate and stable full-field vibration reconstructions than conventional PINNs, particularly under conditions involving high-frequency modes, and highlights the potential of hybrid data-physics neural frameworks as an efficient and reliable approach for solving complex PDE-governed dynamical systems.
Hai-Long Liu, S. Hedayatrasa, Yunpeng Zhu et al.· e-Journal of Nondestructive...· 0 citations
TRACE (Math&Lienhart, arXiv:2602.01135) reads causal graphs over event types out of a pretrained autoregressive sequence model by thresholding a per-position conditional-mutual-information estimate at a fixed tau. We independently replicate its headline synthetic result: with tau selected on a validation split, mean per-sequence F1 against exact interventional truth reaches 0.90-0.91 at vocabulary size 1000 (paper: 0.91) and 0.86-0.91 from 100 to 2000. First, the optimal threshold is pinned to the truth margin, not to any constant: at every size the errors at tau* straddle the delta = 0.05 margin defining ground truth (missed true edges lie just above it, accepted false ones just below), and the blind optimum lands near delta/2 times the estimator's calibration, confirmed out of sample at 5000. Second, at a single global threshold TRACE mostly recovers a direct, adjacent-influence graph: lag-1 true edges are recalled at 0.97-0.99, while true edges at lag 2 or more read orders of magnitude lower---the reading-scale price of randomizing mediating positions, which an exact test of direct causal effect requires when the truth is unknown. A per-lag threshold family recovers a third to a half of lag-2 truth; on lag-uniform data one validated threshold recalls every lag at 0.40-0.87, 8-26 pp below an atomic-intervention control at lags 3-6. Third, the default lag decay of the paper's synthetic benchmark concentrates about 85% of interventional truth at lag 1 and pushes the rest below the estimator's noise floor, so headline F1 there certifies lag-1 recovery only and conflates the benchmark's skew with the algorithm's own limit; a flatter decay separates the two. Fourth, F1 saturates from N = 2 particles at the selected threshold---a property of the threshold's margin over the noise floor, not of the estimator, which converges as N^(-1/2). We distill five practitioner rules.
A.V. Chadyuk, Alicia Zhang, Roy Kucukates· 0 citations
This work systematically compares two state-of-the-art frameworks-Physics-Informed Neural Networks (PINNs) and Optimizing a Discrete Loss (ODIL) across benchmark elliptic, hyperbolic, and parabolic problems, culminating in a challenging inverse source reconstruction task.
Vasco L. Carvalho, J. C. F. Pereira· Computation· 0 citations
We propose a conformal prediction framework for quantifying the error of physics-based predictors used in control, where simple models are preferred for synthesis, certification, and real-time use. Because these models are selected for compatibility with the intended application rather than for maximal predictive accuracy, their error combines process noise with a state-dependent discrepancy. A data-driven discrepancy estimate defines an asymmetric nonconformity score: errors consistent with the learned discrepancy are penalized less than equally large in the opposite direction. The sets remain in the nominal model's error coordinates and are physics-consistent, i.e., they contain a ball at the origin. The construction is agnostic to the discrepancy model (kernel, neural-network, or other), preserves finite-sample marginal validity under exchangeability, and provably narrows the interval over a characterizable state-input region. We further show that, for RKHS models, the power function provides a local confidence measure for adaptive score design and we extend the construction to the multivariate case via a Minkowski-gauge score yielding a jointly calibrated disturbance set.
Cesare Donati, F. Dabbene, Martina Mammarella· arXiv.org· 0 citations
A Physics-Informed Error Field Learning (PIEFL) framework for PINNs is proposed, which avoids continuous optimization of the entire solution space and focuses computational resources on correcting existing prediction errors.
Physics-informed neural networks (PINNs) approximate partial differential equations (PDEs) by enforcing governing equations and boundary conditions during training, but their accuracy depends on how collocation points are distributed and updated. We propose spatiotemporal compositional active sampling (STCAS), a reference-assisted offline configuration procedure that uses an analytic or high-accuracy numerical solution to rank complete three-stage sampling plans. It screens eight fixed rules, forms a task-specific shortlist, and evaluates bounded fixed, switched, and locally blended plans with independent selection sets and a composition guard. A safety-anchor decision retains the standard PINN unless the selected candidate is at least 5% better. Across five evaluations on 18 analytically specified two-dimensional Poisson tasks, this protocol improves 16 task means and ties two, reducing aggregate relative-L2 error by 12.8% (hierarchical-bootstrap 95% interval [6.78%,19.32%]; one-sided paired Wilcoxon p=2.19×10−4). Against the confirmed fixed plan, aggregate error decreases by 8.2%. In comparison experiments designed for two transfer tasks and matched for main PINN training budgets, STCAS achieves the lowest aggregate mean reported error among the compared methods for both a steady convection–diffusion equation and a nonlinear time-dependent Burgers equation; its offline search cost is additional.
Ju-Zheng Zhang, Shi-Yang Li, Tao Zhu et al.· Mathematics· 0 citations
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