VATO: A Vortex-Force-Aware Transformer Operator for Unsteady Separated Aerofoil Flows

Accurate prediction of unsteady separated flows is challenging because the aerodynamic loads depend on nonlinear separation and vortex-shedding dynamics. Although high-fidelity CFD resolves these mechanisms, its cost limits repeated use in design and control. Standard field-level surrogate training, however, does not distinguish the flow regions that contribute most strongly to the aerodynamic loads. We introduce VATO (Vortex-Force-Aware Transformer Operator), which couples the Vortex Force Map (VFM) method to a geometry-aware neural operator through two complementary mechanisms. VATO-S adds training-only supervision of the local VFM force-contribution field, with no increase in model size or inference cost. VATO-A uses VFM contribution and sensitivity fields to prioritise force-relevant source locations for residual cross attention. The methods are evaluated on unsteady CFD data for double-edged-plate aerofoils over 54 trajectories from nine geometries. Over lead times of 1-20~ms, VATO-S reduces velocity, pressure, and vorticity errors by 10.4\%, 1.0\%, and 15.6\%, respectively, while VATO-A achieves reductions of 15.8\%, 7.5\%, and 31.2\%. VATO-S gives the lowest VFM-derived drag error, whereas VATO-A gives the lowest pressure-derived lift and drag errors. Over lead times extending 50\% beyond the training range, VATO-A retains a 26.9\% reduction in vorticity error and larger improvements in all four force readouts, despite reduced gains in velocity and pressure. These results show that force-aware operator learning can improve both flow-field prediction and aerodynamic functional accuracy in unsteady separated flows.

A Constitutive Markov Physics-Informed Neural Operator (MPNO) for Autoregressive Stability in Transient Dynamics

Neural operators applied to transient-dynamics PDEs with strong discontinuities exhibit autoregressive instability: in concrete-penetration stress-field prediction, the wavelet neural operator (WNO) diverges in autoregressive rollout, while MeshGraphNets collapse to zero predictions. WNO's instability stems from the lack of a structural constraint on the spectral radius of its propagation operator; the Fourier neural operator (FNO) is stable in these measurements but only emergently, not by construction. We propose a constitutive Markov physics-informed neural operator (MPNO) modeling one-step evolution as a Markov (row-stochastic) propagation operator. Physics-coupled edge weights (acoustic-impedance harmonic mean, contact area, and traction amplitude) encode material-interface constitutive information into a nonnegative symmetric adjacency matrix W; after normalizing the graph Laplacian L = D - W by lambda_max, the propagator P = I - alpha*L~ is constructively constrained to spectral radius rho(P) <= 1, suppressing exponential amplification of autoregressive errors. Stability is thus a designable architectural property, not an optimized loss objective. On three PDEs (Burgers and two-dimensional transverse-section concrete penetration), MPNO rolls out stably with bounded error on all test seeds at 100/135/165 m/s; the single-step relative L2 error is 0.7304 +/- 0.0008, better than WNO and comparable to FNO at about one quarter of FNO's parameters. The edge-weight formula transfers across scenarios by replacing material-property variables. With about 20K parameters, MPNO delivers roughly 10^5x inference speedup over LS-DYNA.

Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations

In this paper, we introduce the Inertial Manifold Neural Operator (IMNO) for solving dissipative time-dependent partial differential equations (PDEs). The long-time dynamics of such systems often exhibit an effective low-dimensional structure due to dissipation. Unlike standard neural operator architectures such as the Fourier Neural Operator (FNO), IMNO explicitly leverages the low-dimensional structure to achieve better physical interpretability, accuracy, and stability in long-horizon autoregressive training and prediction for nonlinear dissipative PDEs. For shift-equivariant PDEs, we further introduce a shift-equivariant variant (IMNO-SE) of the proposed neural operator, ensuring that a spatial shift in the input induces the same spatial shift in the output. This symmetry-preserving inductive bias substantially improves its performance in shift-equivariant PDEs. Extensive benchmark experiments are presented to evaluate IMNO's performance numerically.

KPI-Conditioned Generative Design of Automotive Hood Inner Panels: A Two-Stage Retrieval-Generation Pipeline with Surrogate-Based Performance Estimation

An inner hood panel must meet a deflection target, stay below a stress limit, and hit a mass target. Machine-learned surrogates have made the forward direction, geometry to performance, fast and routine. The inverse direction, producing geometry from a stated requirement, remains largely unaddressed for industrial parts whose design space is organized into discrete topology families rather than a continuous parameterization. This work presents a two-stage pipeline for that inverse problem. A reachability stage determines which topology families can satisfy a given requirement vector. A conditional variational autoencoder then generates point-cloud geometry within a selected family, and a neural-operator surrogate estimates the performance of each candidate. The pipeline is built entirely from public data and freely available compute, and is deployed as an interactive tool. The pipeline works, with qualifications that are reported as primary findings rather than caveats. The surrogate is accurate in aggregate, but its error is comparable to the performance differences it is asked to discriminate, which bounds what can be claimed for any individual generated design. That ratio of surrogate error to within-class signal is argued to be the quantity that determines whether a pipeline of this kind can work at all.

GCNO: Gramian Chebyshev Neural Operator for Physics-Based Compression of Wireless Channels

Large antenna arrays allow wireless systems to serve more users and achieve higher data rates, but they also make channel feedback expensive: the receiving device must repeatedly report a large complex-valued channel matrix to the base station. Most neural compressors treat this matrix like an image and replace it with a fixed-length code that only a matched neural decoder can interpret. The message therefore does not adapt to channel complexity, and changing the antenna count typically requires retraining. We ask whether a device can instead report only the few dominant propagation paths underlying each channel. We introduce the Gramian Chebyshev Neural Operator (GCNO), a physics-based, variable-rate compressor that identifies a sample-dependent set of path directions. GCNO uses receive-transmit channel structure to locate paths, a first-order Taylor correction to refine directions that fall between grid points, and least squares to recover their complex strengths. It is trained without path labels, and the base station reconstructs the channel analytically from the transmitted path tuples rather than through a learned decoder. Across three ray-traced environments, GCNO achieves better reconstruction accuracy at the same payload - or lower payload at the same accuracy - than neural feedback baselines, and transfers to unseen antenna counts without retraining.

Physics-Unrolled Neural Operator for Wireless Field Modeling

Radio maps are essential for wireless decision-making tasks such as access-point placement, coverage planning, and localization, but their fine spatial details are governed by complex propagation effects and are costly to simulate accurately. Machine learning offers a path to high-fidelity radio-map prediction without running expensive high-fidelity simulations for every scene. However, generating high-quality training labels at scale is also difficult: the affordable labels come from finite-ray simulations, which are richer than low-fidelity inputs but carry residual Monte Carlo noise. We address this challenge with Physics-Unrolled Hybrid Neural Operator (PU-HNO), a three-stage cascade that predicts high-fidelity indoor radio maps from low-fidelity ray-tracing outputs and scene priors by progressively capturing reflection, diffraction, and scattering effects, rather than treating radio maps as generic images. We prove that, under conditionally unbiased label noise, the model can learn stable propagation structure and outperform its own training labels. Experiments across diverse floorplans show that PU-HNO outperforms image-to-image baselines, wireless learning models, and monolithic neural operators across both image-quality and wireless deployment metrics.

Iterative Refinement Diffusion for Super-Resolved Data Assimilation of Multiscale Physical Systems

Recovering high-resolution states from sparse, low-resolution observations is a central challenge in scientific machine learning and data assimilation. Classical data assimilation exploits temporal information through forecast-analysis cycles, but often requires repeated access to expensive high-resolution forecast models. Generative super-resolution can recover unresolved structure from coarse observations, but is commonly used as a one-shot mapping that does not fully exploit constraints from past states. We introduce Iterative Refinement (IR), a learned data assimilation framework that combines these perspectives. Instead of performing a single coarse-to-fine reconstruction, IR decomposes the task into resolution-wise forecast-analysis operations across a multiresolution hierarchy. At each stage, a shared neural operator with resolution-dependent spectral mode slicing provides a dynamical prior, while a shared conditional diffusion corrector uses the current coarser-resolution state to produce a refined posterior at the next finer resolution. We evaluate IR on one-dimensional stochastically forced Burgers dynamics and two-dimensional Kraichnan turbulence. On the challenging 256x256 Kraichnan benchmark, IR achieves an RMSE of 0.184 and an SSIM of 0.836, outperforming spectral upsampling, one-shot diffusion super-resolution, enhanced deep super-resolution, and an autoregressive forecaster. On the more constrained Burgers testbed, IR remains competitive with one-shot diffusion, which achieves the lowest RMSE. These results show that one-shot generative reconstruction can be effective for simpler settings, while hierarchical forecast-analysis refinement becomes advantageous in strongly multiscale and underdetermined regimes. Overall, IR combines temporal priors, generative correction, and multiresolution reconstruction for learned data assimilation in complex physical systems.

A Physics-Informed Hybrid Neural Operator for Transient Magnetization Prediction in Power Magnetics

Magnetic components in high-frequency, high-power-density converters are increasingly driven by non-sinusoidal flux-density waveforms with fast transitions, minor-loop operation, dc bias, and temperature variation. Under these conditions, steady-state core-loss formulas and single-valued material curves cannot fully capture transient magnetization responses. This work proposes the Physics-Informed Hybrid Neural Operator (PI-HNO), a compact material-specific neural model with B-H energy-consistency regularization for core-loss-oriented transient magnetization prediction. Given the measured B(t)-H(t) history, the input B(t) series over the prediction interval and operating-condition information, PI-HNO predicts the H(t) series and the corresponding reconstructed B-H trajectory. The model integrates a local recurrent branch for boundary-state representation and rate-dependent response evolution with a Preisach-inspired global branch that extracts waveform-level hysteresis context. Evaluation on the MagNetX transient database using material-specific models for 14 ferrite materials demonstrates that PI-HNO achieves a compact trade-off between sequence accuracy and B(t)-H(t) energy consistency, with the mean and 95th percentile B(t)-H(t) energy consistency errors of 1.92% and 7.60%, respectively, using only 4777 trainable parameters per model. Ablation studies further demonstrate that the local, global, and energy-aware regularized components provide distinct contributions to transient magnetization prediction.

Amortizing the Calibration Triple: A Projection-Consistent Neural Operator for Local-Stochastic Volatility

Local-stochastic volatility (LSV) combines vanilla marginals with richer smile dynamics, but calibration requires a slow, noisy and sequential McKean--Vlasov fixed point. We learn a projection-consistent operator for the calibration triple. Given finite quotes and a stochastic-volatility (SV) backbone, it jointly returns an implied-volatility surface subject to static-arbitrage constraints, its Dupire local volatility, LSV leverage and the conditional moment required by the projection identity. Starting from option-price marginals, we derive a division-free Dupire residual in log-implied-variance coordinates and a quotient Fokker--Planck equation after Gyöngy projection. Deep Operator Network (DeepONet) and Fourier Neural Operator (FNO) implementations enforce quote fit, static-arbitrage, Dupire and projection constraints. For the witness-augmented residual system, we prove conditional identification and empirical consistency under LSV existence and inverse residual stability. In controlled synthetic tests, forward-start and cliquet errors differ from a particle method by 0.1 and 0.2 percentage points, while calibration latency falls from 98.5 to 0.6 ms. Compared with the tested baselines, local-volatility root-mean-square error (RMSE) falls by 36% and leverage RMSE by 7-16%. These results support amortizing the LSV fixed point: the expensive solve moves offline, while online calibration reduces to a single projection-consistent operator evaluation.

A Physics-Informed Neural Operator for Thermal Ranking of Low-Cost Wall Materials in Hot-Dry Climates

Identifying cost-effective indigenous building materials that minimise heat penetration through walls is critical for indoor thermal comfort in low-income rural housing in hot-dry climates, where summer temperatures routinely exceed 45 C. We present a two-stage computational framework for thermal ranking of five low-cost indigenous wall materials: mud brick, clay-straw adobe, lime-stabilised bamboo panel, fired clay brick, and lime-mud composite. First, a validated Crank-Nicolson finite difference method (FDM) solves the one-dimensional transient heat equation with Robin boundary conditions under diurnal solar and outdoor air-temperature forcing, generating 1500 periodic-day solutions across a nine-dimensional parameter space by Latin Hypercube sampling. Second, a Physics-Informed Neural Operator (PINO) with a Fourier Neural Operator (FNO) backbone learns the parameter-to-solution operator mu -> T(x,t), enforcing both data fidelity and PDE consistency. The trained PINO attains a relative L2 field error of 5.14e-4 and a 0.201 K mean absolute error on the peak inner surface temperature, preserving the FDM material ranking exactly; PINO trained on 150 FDM samples matches a data-only FNO trained on twice as many, so the physics loss is most valuable when data are scarce. The periodic-day formulation also yields the ISO 13786 time lag and decrement factor, reproduced to within 0.99 h and 0.010. At nominal hot-dry summer conditions, clay-straw adobe achieves the best cost-performance index among widely available materials. A climate sweep, confirmed by FDM spot checks, reveals a regime boundary: under sub-ambient outdoor conditions the ranking inverts to conductive fired clay brick, delineating heat-exclusion and heat-rejection regimes. The framework supports evidence-based material selection for post-flood reconstruction in hot-dry regions.

PIKFNO: An Interpretable Neural Operator Based on Physics Informed Kernel Function

This work proposes a new interpretable neural operator framework, termed the Physics Informed Kernel Function Neural Operator (PIKFNO), which explicitly incorporates physics informed kernel functions derived from governing equations into the neural operator architecture. Unlike traditional neural operators such as DeepONet, which rely on deep networks to implicitly learn basis functions, PIKFNO constrains the trunk network through physics informed kernel functions, thereby aligning its operator structure with the kernel expansions used in meshless collocation methods. Two construction strategies are introduced: one learns kernel functions directly from data, where the learned kernel can be regarded as a nonsingular fundamental solution, while the other builds them through transformations of analytical fundamental solutions. Numerical experiments demonstrate that PIKFNO achieves high predictive accuracy with substantially improved interpretability and superior generalization under limited training data. The proposed framework offers a new pathway for developing efficient, physically consistent, and interpretable neural operators.

Neural Operator-enabled Topology-informed Evolutionary Strategy for PDE-Constrained Optimization

The inverse design of physical systems governed by partial differential equations is computationally demanding due to the high dimensionality and non-convexity of design spaces. Generative models for inverse design often lack robustness and transferability, whereas evolutionary strategies are robust but struggle in high-dimensional spaces. This paper introduces a Neural Operator-enabled Topology-informed Evolutionary Strategy (NOTES) that integrates dimensionality reduction, representation learning, and evolutionary optimization for efficient and transferable inverse design. NOTES couples a DeepONet-based neural operator with the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) to perform global optimization in a compact latent space that encodes topology-aware priors while discovering high-performance designs for unseen operating conditions. Applied to nanophotonic beam-deflector inverse design governed by Maxwell's equations, NOTES reduces the design dimensionality from 256 to 25 and consistently achieves over 95 percent efficiency, outperforming CMA-ES, topology optimization, and other baselines. Applied to structural optimization, NOTES discovers designs that achieve compliance down to 246. By decoupling topology learning of a DeepONet from the governing physics in a PDE solver, NOTES provides a flexible and transferable framework for the inverse design of physical systems.

Knowledge-Constrained Shape Optimization with a Mixture-of-Experts Neural Operator for High-Confidence Design

Engineering shape optimization faces challenges in both expert-dependent problem setup and surrogate-model reliability. In practical aerodynamic design, optimization settings such as editable regions, deformation ranges, and design-preservation constraints are typically specified manually by experienced engineers, while surrogate-based optimization may become unreliable for heterogeneous geometry databases and out-of-distribution designs. To address these challenges, we propose a knowledge-constrained shape-optimization framework that translates knowledge-based constraints and user intent into quantifiable parameters of DFFD-based deformation operators, enabling engineering-aware and controllable constrained optimization. We further develop a Mixture-of-Experts Neural Operator (MoE-NO) to improve drag prediction and trend consistency over heterogeneous aerodynamic datasets. Based on the MoE-NO encoder and Mahalanobis distance, an uncertainty-estimation strategy is introduced to detect out-of-distribution geometries and selectively trigger physics-solver feedback for local sample enrichment. Experiments on in-house MPV, SUV, and Sedan datasets show that MoE-NO achieves a test-set MAPE of $1.16\%$ and a trend-prediction accuracy of $94.34\%$, outperforming the best baseline results of $1.52\%$ and $90.34\%$, respectively. Vehicle shape-optimization experiments further yield CFD-validated drag coefficient reductions of approximately $4\%$ to $10\%$.

Quadrature-Aware Complex-Linear Neural Operator for Boundary-to-Field Prediction in Resonant Acoustics

Repeated prediction of acoustic fields from spatially distributed boundary excitation is computationally expensive when each source realization requires a new wave simulation. This work introduces a quadrature-aware complex-linear boundary operator (CLBO) that maps complex normal velocity on a vibrating surface to complex pressure at receiver locations. The model couples learned source and receiver basis functions through an explicit complex surface-quadrature contraction, so the boundary excitation enters linearly by construction. This preserves complex superposition, homogeneity, and zero response to zero excitation, while representing the source through coordinates, normals, and quadrature weights rather than a fixed flattened input vector. Reference data were generated using a verified three-dimensional multiple-relaxation-time (MRT) lattice Boltzmann solver and stored in a solver-agnostic boundary-to-field format. CLBO was compared with a fixed-sensor complex DeepONet under matched case splits and optimization settings, with additional tests of structural consistency, receiver-coordinate interpolation, source discretization, source-family holdout, label efficiency, physics-informed ablations, unseen source mixtures, and computational cost. Across five training seeds, CLBO achieved a mean complex relative field error of 0.184 +/- 0.00771, compared with 0.367 +/- 0.00742 for DeepONet. Its measured source-superposition error was 1.31 x 10^-7, and its mean error on newly simulated mixed-source cases was 0.237, compared with 0.415 for DeepONet. Inference was 1.83 x 10^4 faster than the reference calculation for the reported query size. These results show that enforcing the known complex-linear boundary-to-field structure improves physical consistency and generalization under distributed acoustic excitation.

Gradient-based inverse lithography for EUV masks via the waveguide method and a physics-informed neural operator

Gradient-based inverse lithography technology~(ILT) for extreme ultraviolet~(EUV) masks is presented. A novel framework treats the differentiable waveguide method and the recently proposed waveguide neural operator~(WGNO) as end-to-end physics engines, recovering the permittivity of the absorber of the mask through automatic differentiation of the full forward diffraction model. Numerical experiments on realistic 2D and 3D absorbers of the mask (TaBN, La, U) at $λ{=}11.2$~nm show that the considered ILT methods make it possible to obtain a mask structure that achieves the desired field on the wafer.

A fast direct solver based neural network for solving PDEs

The matrices arising from large scale $N$-body problems can be efficiently represented using hierarchical matrices, whose key idea is that the admissible off-diagonal sub-matrices can be well approximated by low-rank matrices across a hierarchy of matrix partitions. HODLR (Hierarchical Off-Diagonal Low-Rank) matrices are a subclass of hierarchical matrices in which all off-diagonal submatrices at every level of a recursive binary partition are low-rank. In this article, we present a neural network that learns the inverse operation of HODLR matrices based on the fast direct solver for HODLR matrices developed by Ambikasaran and Darve (2013). We further extend the architecture to learn nonlinear solution operators associated with PDEs by replacing some of the linear layers with deep sub-networks. We demonstrate the performance of the proposed architecture by performing a comprehensive set of experiments that include (i) solving a linear problem such as the Fredholm integral equation of the second kind, (ii) solving PDEs such as the nonlinear Schrödinger equation, Burgers' equation, and the steady-state Darcy's flow equation, (iii) generalization study across varying parameter values, (iv) comparing the inference time of the proposed network with the run time of a classical numerical solver, and (v) comparing the proposed network with some of the existing neural operator learning networks.

MR-GVNO: A Geometry-Aware Variational Physics-Informed Neural Operator for Mindlin-Reissner Plates on Irregular Domains

Plate and shell structures are widely used in engineering, making rapid response prediction under varying geometries, materials, and loads highly desirable. However, conventional finite element methods require repeated modeling and solution, resulting in high computational costs. This study proposes a geometry-aware variational neural operator for Mindlin-Reissner plate problems, termed MR-GVNO. The method uses boundary point clouds to represent irregular geometries and employs separate encoders for spatially varying material fields, pressure loads, and scalar physical parameters. A cross-attention mechanism integrates these inputs with query point information to predict transverse deflections and rotations at arbitrary locations. MR-GVNO is trained without labeled solution data using a variational physics-informed loss derived from the discretized total potential energy. It directly processes irregular point clouds and allows different physical fields to be discretized independently, avoiding interpolation onto a common grid. Numerical experiments on single-hole, double-hole, and L-shaped plates demonstrate accurate response prediction under homogeneous and heterogeneous materials and uniform and random loads. The model also achieves millisecond-level full-field inference and favorable cross-geometry generalization.

Towards Data-Efficient Cross-Device Generalization of Grad-Shafranov Equilibria via Transfer Learning Neural Operator

Real-time reconstruction of magnetohydrodynamic equilibria is essential for plasma shaping, stability assessment and feedback control in magnetic confinement fusion. However, Grad-Shafranov equilibrium calculations remain largely device-specific and iterative, limiting their use in latency-constrained control settings. Existing neural approaches can accelerate individual equilibrium predictions, but they do not generally provide reusable models across changing plasma boundaries or tokamak geometries. Here we show that equilibrium reconstruction can be recast as a cross-device operator learning problem. We develop a domain-specific neural operator framework that maps geometry and profile parameters directly to the poloidal flux field, replacing repeated solve-on-demand computation with amortized operator inference. Using the analytically tractable Solov'ev family as a controlled Grad-Shafranov testbed, we generate equilibria across eight geometrically distinct tokamak-like configurations and benchmark five neural operator architectures under four transfer-learning strategies. Single-geometry pretraining gives poor transfer to unseen devices, whereas multi-geometry pretraining enables data-efficient adaptation. The Wavelet Neural Operator gives the strongest cross-geometry performance, reaching mean relative L2 errors below 4% with 100 labelled target equilibria and below 2% with full fine-tuning. The predicted magnetic fields satisfy the divergence-free constraint to numerical precision, and four architectures achieve millisecond or sub-millisecond inference. These results identify neural operator pretraining as a route towards reusable, real-time equilibrium inference across fusion device configurations.

A Geometry-Aware Triplane Field Network for Vehicle Aerodynamic Prediction

High-fidelity computational fluid dynamics (CFD) is crucial to vehicle aerodynamic analysis, but its cost still constrains early-stage design exploration. Machine-learning-based surface-field prediction offers a faster alternative if the model can efficiently capture both global flow context and local geometric detail. This work proposes a machine-learning-based method, named the geometry-aware triplane field network (GTF-Net), for vehicle aerodynamic pressure and wall shear stress prediction. GTF-Net constructs triplane features directly from sampled surface points through a shared multilayer perceptron (MLP) and smooth bilinear rasterization. The planes are then processed by a dual-stream backbone that combines adaptive Fourier neural operator (AFNO) spectral mixing with convolutional neural network (CNN) refinement, so long-range aerodynamic coupling and local geometry-induced variations are modeled in the same representation. At query stage, sampled triplane features are combined with vehicle-aligned directional coordinates, normal-projection features, and a voxel-based curvature proxy. GTF-Net is compared with Transolver, geometry-informed neural operator (GINO), and TripNet, a triplane-based surrogate model. GTF-Net improves the relative L2 error from the strongest baseline value of 0.157 to 0.145 for pressure prediction and from 0.237 to 0.226 for wall shear stress prediction. Ablation results show that AFNO mixing, local CNN refinement, and query-side geometric encoding each contribute to accuracy, supporting the proposed mechanism of combining structured triplane representation with explicit aerodynamic geometry cues.

Will Accurate Fields Mislead Photonic Design? FromGlobal Accuracy to Port Readout

Neural field surrogates can accelerate photonic design loops, but a surrogate that looks accurate in global field error can still mis-rank candidate devices when the final decision depends on localized output-port readouts. This risk is acute in propagation-dominated MMI splitters and couplers, where port power, splitting, phase, and coupling are determined by accumulated modal interference and output-window aggregation rather than by average field similarity alone. We study this field-to-design mismatch through a Field/Mediator/Readout view that separates dense complex-field error from propagation-profile and output-window errors before port aggregation. To align the surrogate with this chain, we propose PaNO, a propagation-aligned neural operator that keeps the full-field prediction interface while organizing latent states around local boundary structure, transverse modal content, axial propagation, and cross-mode interaction. We also evaluate PaNO-R2, an output-aware feedback variant for residual field components near the port region. On a 15-wavelength tunable $3{\times}3$ MMI benchmark with 4608 held-out fields, PaNO lowers NeurOLight's port-power error from 0.2018 to 0.0739 despite slightly higher cMAE, showing that global field accuracy alone is not sufficient for design-relevant readout fidelity. PaNO-R2 attains the best cMAE, propagation-profile error, output-profile error, and port-power error, reducing NeurOLight's port-power and output-profile errors by 72.7\% and 72.5\%.