Abstract KPFM measurements frequently suffer from variable signal-to-noise ratios due to suboptimal acquisition conditions, particularly in time-sensitive experiments. While conventional filtering can reduce noise, improper parameter selection risks distorting quantitative surface potential values, and manual tuning by experts limits reproducibility. Here, we propose a self-supervised deep learning framework for automatic filter parameter optimization tailored to KPFM data.The architecture combines an MLP pathway that predicts global filter parameters from 22 image-level features and structural complexity information, with a CNN pathway that generates pixel-wise local parameter adjustments. Trained on 13,218 real-world KPFM data without requiring clean reference images, the model adaptively modulates filter strength according to image noise level: achieving effective noise reduction with near-perfect edge preservation (EPI = 0.984) in high-noise conditions, while minimizing data alteration (PSNR = 39.4 dB, SSIM = 0.940) in low-noise conditions. The framework supports five classical filter types (Wiener, Gaussian, Mean, Bilateral, and Total Variation), providing a reproducible and objective alternative to manual parameter tuning.
Abstract The interphase formed between solid electrolytes and alkali metal anodes determines whether decomposition is self-passivating, solid electrolyte interphase (SEI) or continuously propagating, mixed conducting interphase (MCI). We employ machine learning interatomic potential molecular dynamics to simulate the Na 3 PS 4 /Na interface at scales of 500,000 atoms and 10 ns. Three features distinguish this system from the passivating Li 7 P 3 S 11 /Li interface: (1) P–P transport correlations enabling cooperative phosphorus migration toward the anode, (2) amorphous Na 2 S domains lacking nanocrystalline order, and (3) persistent Na–P connectivity pathways spanning the interphase from the onset of decomposition. These characteristics are consistent with mixed ionic-electronic conducting behavior and inconsistent with self-limiting SEI character. While definitive classification requires direct electronic conductivity measurements of the amorphous interphase, this work establishes atomistic mechanistic criteria for interphase classification and demonstrates that MLIP simulations can access the length and time scales necessary to resolve transport mechanisms at solid electrolyte interfaces.
The exact exchange-correlation potential $v_{\mathrm{xc}}$ in time-dependent density functional theory is a causal functional of the full density history, a dependence that widely used adiabatic approximations discard. We ask whether this history dependence can be approximated by a learnable finite-dimensional memory state while retaining a Hamiltonian-consistent description of the Kohn–Sham phase-space dynamics. To this end, we propose a Three-Head neural architecture consisting of a Hamiltonian dynamics head that induces a symplectic vector field from a parameterized scalar energy, a direct vector field head that supplies a flexible correction pathway, and a memory XC potential head that reconstructs $v_{\mathrm{xc}}$ from density history through a recurrent memory module with cross-window carry. We evaluate the model on two physically distinct benchmarks. On the one-dimensional two-electron electron–hydrogen scattering model, it attains the lowest density reconstruction error among all baselines over the full propagation window, by the largest margin during the pre-collision approach phase, and is the only model that recovers the peak-and-step structure of $v_{\mathrm{xc}}$ in the subsequent strongly nonadiabatic scattering window. On a one-dimensional helium Rabi oscillation model under strong-field driving, it reduces the correlation-potential reconstruction error by roughly an order of magnitude relative to all adiabatic baselines at every driven phase of both the Rabi and optical cycles, including the adiabatically exact ground-state limit, which shows that the residual error along the adiabatic axis is controlled by the missing density history rather than by the accuracy of the time-local map. Ablation studies show that structured dynamics and memory-based potential learning are mutually supportive rather than separable.
Abstract Despite rapid progress in retrieval-augmented generation (RAG), existing systems still struggle to retrieve both fine-grained entity-level evidence and higher-level abstract knowledge from biomedical literature. Graph-based RAG offers a promising alternative to flat retrieval, and hierarchical Graph RAG extends this idea by enabling coarse-to-fine retrieval across multiple levels of abstraction. Its effectiveness, however, depends on hierarchy construction choices, uneven community connectivity, and query-dependent layer choice. These challenges are particularly important in literature-derived biomedical graphs, where the hierarchy is constructed automatically from documents rather than specified in advance. We introduce HAGRAG, a hierarchical attributed Graph RAG framework for PubMed-derived diabetes literature. HAGRAG constructs multi-level attributed communities, summarizes them across layers, indexes them using hierarchical navigable similarity search, and supports layer-aware filtering during retrieval. It further evaluates alternative community detection algorithms and layer-weighting strategies to determine how hierarchy design affects retrieval behaviour. Across comparisons with baseline systems, HAGRAG achieves the strongest overall performance, including accuracy of 0.80, relevancy of 1.00, recall of 0.54, and mean semantic similarity of 0.65. Answer quality is further assessed through NLI-based entailment checks and independent diabetes-clinician validation. The results show that Leiden produces the most coherent hierarchy, while retrieval quality remains sensitive to clustering and weighting choices. Overall, the findings show that effective hierarchical Graph RAG depends not only on graph augmentation, but on constructing, navigating, and weighting the hierarchy.
The development of laser-plasma accelerators has brought new opportunities for tumor radiotherapy. After appropriate energy selection and beam transport, laser-accelerated proton beams can provide depth-dose characteristics relevant to proton radiotherapy. Owing to their ultrashort pulse duration and extremely high instantaneous dose rate, they also provide a promising platform for ultra-high-dose-rate and FLASH-related radiobiological studies. However, the low-frequency single-shot mode and dose fluctuations of laser proton beams, resulting from the transient nonlinear acceleration process, limit the accuracy of irradiation doses. Traditional irradiation control methods are inadequate to address these issues. Here, we propose a model-based adaptive dose-delivery method based on sequential decision optimization, which can maintain stable dose control with an accuracy of 5% under unstable laser-accelerated proton beams. This method involves adjusting the beam transport system and employing feedback control systems to monitor and adjust beam parameters in real time, adaptively compensating for upstream instabilities without altering the conditions at the driver end. Simulation results show that these schemes significantly outperform traditional methods in terms of dose delivery accuracy and irradiation efficiency, providing important references for the further development of laser proton radiotherapy technology.
Graphs provide a powerful framework for modeling complex systems, but their structural variability poses significant challenges for analysis and classification. To address these challenges, we introduce Graph Autoencoder Uncovering Descriptive Information (GAUDI), an unsupervised graph deep learning framework designed to capture both local details and global structure. GAUDI employs an hourglass architecture with hierarchical pooling and upsampling layers linked through skip connections, which preserve essential connectivity information throughout the encoding–decoding process. Even though identical or highly similar underlying parameters describing a system’s state can lead to significant variability in graph realizations, GAUDI consistently maps them into nearby regions of a structured and continuous latent space, effectively disentangling invariant process-level features from stochastic noise. We demonstrate GAUDI’s versatility across multiple applications, including small-world networks modeling, characterization of protein assemblies from super-resolution microscopy, analysis of collective motion in the Vicsek model, and identification of age-related changes in brain connectivity. Comparison with related approaches highlights GAUDI’s superior performance in analyzing complex graphs, providing new insights into emergent phenomena across diverse scientific domains.
Physics-informed neural networks (PINNs) are limited by three interlocking bottlenecks, the PINN trilemma : Monte-Carlo collocation, serial automatic differentiation, and a resulting stochastic loss landscape that prevents quasi-Newton optimisation. These bottlenecks place an empirical $\mathcal{O}(10^{-2})$ accuracy floor on stiff multiphysics problems such as electrochemical transport, well above the sub-percent precision required for quantitative parameter inference. In this paper we address that accuracy floor by replacing the random-collocation pipeline with a spectral one. We introduce the neural spectral element method (NSEM), which evaluates each network only at fixed Legendre–Gauss–Lobatto quadrature nodes and replaces all derivative calls with precomputed spectral differentiation matrices. The resulting deterministic loss enables limited-memory BFGS (L-BFGS) to reach residuals of $10^{-9}$ – $10^{-10}$ . A Kosloff–Tal-Ezer coordinate map resolves electrochemical boundary layers; a mesh-free neural mortar framework couples multi-element domains. On the four-example Poisson–Nernst–Planck (PNP) benchmark, NSEM attains $10^{-4}$ – $10^{-7}$ relative pointwise error with two orders of magnitude fewer collocation points than the adaptive-resampling PINN baseline. Both a tanh multilayer perceptron and a basis-aligned Legendre Kolmogorov–Arnold network (KAN) backbone attain spectral accuracy within the same NSEM infrastructure, with the KAN requiring roughly half the Adam steps to enter the L-BFGS basin of attraction on the 1D PNP benchmark.
Complex microstructural pattern formation, such as dendrite growth, occurs across a wide range of materials and plays a crucial role in determining their properties and functional performance. While the phase-field method is a powerful computational approach for modeling microstructure dynamics, its substantial computational cost limits its integration into practical materials design workflows. Here, we introduce a machine-learning framework that employs autoregressive deep surrogates, trained on short trajectories from quantitative phase-field simulations of alloy solidification within limited spatial domains. Once trained, these surrogates accurately predict dendritic evolution over extended length and time scales, achieving speed-ups exceeding two orders of magnitude. We demonstrate the effectiveness of this approach through examples of isothermal growth and directional solidification of a dilute Al–Cu alloy, confirming its capability to predict complex microstructural pattern formation. Quantitative comparisons with phase-field benchmarks reveal excellent agreement in the tip-selection constant, morphological symmetry, and primary spacing evolution, further validating our approach.
This work investigates graph-based deep learning approaches for the identification of rare and non-standard particle signatures in the ATLAS experiment at CERN focusing on dark photons, new particles predicted by beyond standard model theories, as a representative case. Such signatures arise from displaced decays and produce highly heterogeneous and irregular energy deposition patterns in part of the ATLAS detector such as the calorimeter, that are difficult to model with conventional approaches. Graph neural networks provide a natural framework to model the sparse and relational structure of these data. We evaluate a range of architectures, including message-passing networks, attention-based models, and graph Transformers. Building on this, we introduce a graph Transformer augmented with a mixture of experts (MoE) mechanism, designed to better capture the intrinsic heterogeneity of these signatures through modular representations. The models are benchmarked on public, simulated ATLAS calorimeter data, and their behaviour is analysed using physics-motivated observables. The results show that graph-based approaches, and in particular the MoE-enhanced graph Transformer, improve sensitivity to rare and complex signal topologies, while providing insight into the learned representations, and as such offer useful guidance for future applications in particle physics and related domains.
Automated exploration of chemical space through machine learning has become a central strategy in modern drug discovery, where the ability to learn transferable molecular representations directly impacts predictive modeling performance. Recent advances in chemical language models have primarily relied on atom-level string encodings such as SMILES or SELFIES, treating molecules as sequences of individual atoms or characters. However, medicinal chemistry reasoning is often fragment-centric, emphasizing scaffolds, functional groups, and modular substructures rather than isolated atoms. In this study, we introduce FragBERTa, a molecular fragment-aware transformer-based representation language model pretrained using masked language modeling on Sequential Attachment-based Fragment Embedding (SAFE) representations. SAFE decomposes molecules into chemically meaningful fragments while preserving attachment information, enabling the model to learn fragment-level structural patterns within a sequence framework. FragBERTa is pretrained on 11 million unique SAFE sequences extracted from multiple public chemical databases and subsequently finetuned on diverse MoleculeNet benchmarks covering both regression (ESOL, FreeSolv, Lipophilicity, PDBbind) and classification (BACE, BBBP, HIV, Tox21, SIDER) tasks under both random and scaffold splits. Our evaluation shows that FragBERTa achieves competitive performance across tasks, often matching or surpassing prior models. In regression, it demonstrates strong gains in lipophilicity and the best scaffold-split performance on protein-ligand binding affinity within our comparison. In classification, FragBERTa attains the strongest reported random-split ROC-AUC results on BACE, HIV, and Tox21 and the strongest reported scaffold-split ROC-AUC performance on BBBP among the compared baselines, and delivers robust performance across bioactivity, permeability, and toxicity benchmarks, indicating that fragment-based representations effectively capture biologically driven and interaction-sensitive endpoints. Collectively, these findings suggest that fragment-based string representations offer advantages over atom-level representations such as SMILES and SELFIES for scaffold-sensitive and interaction-driven tasks, highlighting the value of fragment-level inductive bias for future molecular foundation models in drug discovery.
Physics-informed machine learning of parametric partial differential equation (PDE) families enables rapid prediction across varying physical conditions, yet the resulting task representations are commonly embedded in latent neural features that are difficult to interpret physically. This raises the question of whether a parametric neural PDE solver can make explicit how physical task parameters reorganize the underlying solution representation. To address this gap, we introduce Meta-Sparse, Physics-based, and partially Interpretable Neural Network (SPINN), which maps task parameters to a shallow RBF model with inspectable coefficients, centers, scales, and directional parameters. We show that, across elliptic, transport, advection–diffusion, variable-coefficient, and nonlinear PDE families, the learned bases adapt to and organize around the dominant physical solution structures, including localized forcing responses, characteristic-aligned transport trajectories, diffusion-broadened space–time corridors, and viscous shock fronts. Meta-SPINN works both as a direct predictor for unseen tasks and as a task-aware initializer for subsequent single-instance residual-guided refinement, providing reusable predictions together with an interpretable visualization of how solution geometry changes across a parameter family.
Donor-based spin qubits in silicon are a promising platform for scalable quantum computing due to their long coherence times and high-fidelity gate operations. A viable path for fabricating donor qubit arrays with atomic precision is scanning tunneling microscopy hydrogen depassivation lithography (STM-HDL). However, the identification of the intermediates of the dissociating precursors has traditionally relied on manual, expert-driven interpretation of scanning tunneling microscopy (STM) images, which severely limits throughput and impedes full automation. Here, we introduce an artificial intelligence-powered framework based on a multimodal YOLO (You Only Look Once) neural network, which performs simultaneous analysis of dual-polarity STM images acquired at positive and negative bias voltages. Our approach enables real-time automated classification of the species after dissociation of precursors and atomic-precision localization of dopant precursors during phosphorus donor qubit fabrication in silicon. The proposed multimodal architecture outperforms single-modality models, achieving 92.1% classification accuracy for surface adsorbates and 76.3% localization accuracy, representing a 30.4% improvement in overall performance. To our knowledge, this work constitutes the first demonstration of multimodal deep learning applied to STM image interpretation, offering a key step toward scalable and autonomous fabrication of atomically engineered qubit arrays in silicon.
Parameter identification in nonlinear dynamical systems is complicated by model-form uncertainty arising from systematic biases that violate the zero-mean error assumption of standard data assimilation methods. Recent neural-network-based approaches learn arbitrary bias corrections online. However, they require careful regularization to ensure unique solutions and carry computational overhead from ensemble propagation and in-situ training. We present a framework that integrates the parametrized-background data-weak formulation with attention-based parameter identification networks. It projects model error onto a dictionary of physically motivated spatial templates rather than learning arbitrary corrections. This provides uniqueness through hard subspace constraints rather than soft regularization penalties, albeit at the cost of restricting the representable bias space to patterns anticipated from domain knowledge. The framework is demonstrated on parameter estimation in the Rijke tube model, achieving robust generalization to out-of-distribution (OOD) bias patterns not seen during training. Ablation studies confirm the importance of state supervision for amplitude-sensitive parameters and temporal bias modeling for phase-sensitive parameters. Performance degrades gradually when bias patterns lie outside the template span. This indicates that the attention encoder extracts parameter information from bias-invariant features rather than relying critically on template-based bias capture. The physics-constrained and universal approximator approaches represent complementary points on the flexibility–efficiency tradeoff, suited to different operational contexts.
Machine learning (ML) is increasingly utilized in molecular dynamics (MD) simulations to investigate complex system properties across disciplines ranging from chemical and physical sciences to engineering. However, these methods often require large datasets for model training and may not fully adhere to physical principles, limiting their scientific explainability. Furthermore, the model training remains highly challenging under sparse data. To overcome these limitations, this paper introduces the Molecular Physics-Informed Neural Network (mPINN) framework, designed to solve Newton’s equations of motion for multi-body atomistic interactions without relying on prior observational time-series data. By directly embedding the governing principles of MDs into the neural network training process, the mPINN promotes physical consistency, ensuring that predicted trajectories adhere to conserved thermodynamic quantities. This methodology replaces unconstrained empirical data-fitting with mathematically bound physical relationships, yielding stable and physically valid performance across continuous temporal domains. Unlike traditional MD engines that rely on step-by-step discrete integration, the mPINN operates within a continuous time framework during each training iteration. This approach effectively alleviates issues related to timestep selection and stability commonly faced in discrete simulations. The results demonstrate that the mPINN architecture functions as a reliable, physics-constrained ML framework capable of delivering high-fidelity trajectory predictions for complex multi-body molecular systems.
In the context of the dynGENIE3 (Anh Huynh-Thu and Geurts 2018 Sci. Rep. 8 3384) approach for inferring regulatory network interactions from time-series data, we show that it is possible to modify that algorithm to significantly enhance its prediction reliability. To quantify the level of reliability, we used ground-zero truths based on simulated datasets generated by the GeneNetWeaver (Schaffter et al 2011 Bioinformatics 27 2263–70) tool. Our work introduces novel methods leveraging time-lagged correlations and estimators of mRNA decay rates, leading to significantly improved driver-target inference. Additionally, a temperature-based rescaling of priors was developed to further enhance prediction reliability. Results demonstrate substantial improvements in performance with a particularly notable increase in area under the precision recall curve scores. These advances underscore the possible gains resulting from incorporating priors into gene regulatory network inference.
This study systematically investigates two multi-fidelity strategies used to train machine-learned force fields (MLFFs)—pre-training/fine-tuning and multi-headed training—and elucidates the mechanisms underpinning their success. For pre-training and fine-tuning, we uncover a log–log linear relationship between pre-trained and fine-tuned accuracies that holds across model architectures, model sizes, and quantum-chemical methods. The success of this approach hinges on the quantity and quality of available pre-training data, and, critically, the inclusion of force labels. We demonstrate that pre-trained representations are inherently method-specific, requiring adaptation of the model backbone during fine-tuning. In contrast, multi-headed models learn method-independent backbone representations, where again the heads’ accuracies are log–log linearly related. Relative to pre-training and fine-tuning, these shared representations marginally reduce model performance in most cases. However, this trade-off is offset by practical advantages: multi-headed training extends naturally to multiple labelling methods and enables partial replacement of expensive labels with cheaper alternatives, paving the way towards cost-efficient universal MLFFs.
Scientific machine learning increasingly uses explainable artificial intelligence (XAI) to interpret environmental prediction models, but highly ranked predictors are not necessarily reliable explanations across spatial, resampling, model, and predictor-correlation uncertainty. Here, we introduce a reliability-calibrated XAI framework for separating predictive importance from explanation reliability in scientific machine learning. The framework combines station-held-out validation, shapley additive explanations (SHAP)-based explanations, station-level bootstrap resampling, rank stability, phase-aware directional consistency, predictor ablation, and multimodel comparison into a reliability performance index (RPI), a bounded empirical reliability score that distinguishes influential predictors from stable explanations. We demonstrate the framework using winter streamflow prediction across the Chesapeake Bay watershed, linking daily discharge observations from 51 U.S. Geological Survey stations during 1990–2024 with time-safe lagged climate-mode predictors. Across tree-based models, station-held-out seasonal predictions achieved moderate spatially transferable skill, with mean cross-validated ${R^2} \approx 0.48$ . The RPI framework identified September Pacific Decadal Oscillation (PDO) conditions preceding winter by approximately three months as the most spatially transferable and explanation-stable predictor. This leading ranking was retained under additive, geometric, minimum-component, and threshold-based alternative RPI formulations. In a controlled grouped benchmark with known predictor roles, RPI recovered both materially important, directionally stable, spatially transferable drivers in the top two positions across all three model classes, whereas standard SHAP importance recovered only one. Phase-aware SHAP diagnostics showed consistent directional effects across PDO phase: positive September PDO states were associated with higher predicted DJF discharge anomalies. Model-independent diagnostics supported this direction. Station-level correlations were predominantly positive, with 90.2% of stations showing positive Spearman correlations between September PDO and winter discharge anomaly. Counterfactuals also predicted higher discharge under positive PDO conditions. These results demonstrate that feature importance alone is insufficient for scientific interpretation and that reliability-calibrated XAI provides a more conservative basis for identifying explanation-stable and physically consistent machine-learning relationships.
Abstract Station-level graph and spatial-attention forecasters often produce overly smooth PM 2.5 fields that underestimate localized pollution gradients. The compound spectral smoothing bias examined here arises from three complementary mechanisms. Message-passing aggregation can act as a graph low-pass filter that attenuates high-frequency (HF) spatial modes. MSE optimization can favor lower HF energy when high-frequency graph Fourier coefficients carry proportionally more conditional uncertainty than low-frequency ones. Neural spectral bias further favors low-frequency components during gradient-based learning. A graph spectral diagnostic framework based on the graph Fourier transform quantifies HF energy deficits. Applied to 72 h PM 2.5 forecasting over 228 stations across seven graph and spatial-attention models plus two graph-free recurrent controls, the framework reveals consistent long-horizon HF attenuation. All non-advection entries reach 0.76 to 0.86 percentage-point deficit at 72 h relative to an annual-mean ground-truth HF ratio of 0.91%, a relative reduction of approximately 83% to 94% of ground-truth HF energy. Three targeted interventions are used as diagnostic stress tests under the shared MSE-selected protocol. Explicit advection degrades accuracy by approximately 4.5% despite a mostly stable effective-CFL audit. A spectral residual bypass worsens point metrics due to spatial displacement. Spectral training penalties leave HF recovery unchanged at approximately 6% to 7% of truth. Together, these stress tests delimit the HF-recovery routes evaluated under the shared MSE-selected protocol. Seasonal stratification shows that winter spatial structure is the most recoverable regime for the non-advection models, consistent with expected differences in atmospheric predictability. These results establish HF deficit as a quantitative diagnostic and benchmarking axis for spatial fidelity in station-level air-quality forecasting.
Protein language models (pLMs) leverage large-scale evolutionary data to generate novel sequences, but steering generation toward desired physicochemical properties without sacrificing diversity remains a major challenge. Existing approaches often induce severe diversity loss or require computationally expensive retraining. We introduce Iterative Lookback Monte Carlo (ILMC), a training-free inference-time sampling strategy that interleaves autoregressive elongation with Metropolis–Hastings refinement to approximate sampling from a maximum-entropy target distribution balancing generative quality and steering objectives. We show theoretically that this target distribution is entropy-maximizing under fixed generative quality and steering constraints, and empirically that ILMC produces more diverse samples than standard autoregressive baselines at matched generative quality. Using simple steering potentials, ILMC improves desired molecular properties, yielding higher predicted melting temperatures than both unsteered autoregressive sampling and compute-matched rejection sampling at matched generative quality. ILMC naturally applies to classifier-guided steering, where it outperforms purely autoregressive guidance in diversity while maintaining comparable enrichment of target properties. We validate ILMC on family-specific pLMs and on the multi-family model ProGen3.
We present a systematic ablation study of physics-informed neural networks (PINNs) for level-set advection across four benchmarks of increasing complexity: linear translation (TR), solid-body rotation (RO), reversed vortex deformation (RV), and the Zalesak rotating slotted disc (ZD), covering 69 experiments. For TR, a step learning rate scheduler (StepLR) with eikonal weight $w_\text{eik} = 1.0$ is optimal (mean $L^2$ error $\overline{\mathcal{E}}_{L^2} = 2.09\times10^{-4}$ ). For RO, cosine annealing (CosineAnnealing, minimum learning rate $\eta_\text{min} = 10^{-5}$ ) outperforms StepLR, establishing that scheduler choice is benchmark-specific and cannot be transferred. For RV, reducing $w_\text{eik}$ from $1.0$ to $10^{-4}$ yields an $82\times$ improvement ( $\overline{\mathcal{E}}_{L^2} = 1.51\times10^{-3}$ , mean relative $L^2$ error $\overline{\mathcal{E}}^{\,\text{rel}}_{L^2} = 0.43\%$ , final time $T = 2$ ); extending to $T = 8$ with causal weighting and residual-based adaptive distribution and refinement (RAD + RAR) achieves $0.63\%$ with a standard tanh network, outperforming the PirateNet state-of-the-art (Sota) of Mullins et al (2025) ( $0.85\%$ ). For ZD, four studies are conducted: the eikonal weight study (S1), the progressive sampling study (S2), the architecture study (S3), and the adaptive sampling study (S4). Random Fourier feature (RFF) encoding (bandwidth $\sigma = 2$ ) with $w_\text{eik} = 10^{-2}$ and causal weighting achieves $\overline{\mathcal{E}}_{L^2} = 5.74\times10^{-4}$ in the architecture study (S3), a $10\times$ reduction over the tanh baseline; adding RAD + RAR with RFF $\sigma = 5$ and $M = 32$ causal chunks achieves the best overall ZD result: $\overline{\mathcal{E}}_{L^2} = 4.64\times10^{-4}$ ( $\overline{\mathcal{E}}^{\,\text{rel}}_{L^2} = 0.13\%$ ) in 15.2 min in the adaptive sampling study (S4), outperforming the published Sota. A key finding is an RFF–eikonal joint design constraint: at low bandwidth ( $\sigma = 2$ ), weak eikonal regularization ( $w_\text{eik} = 10^{-4}$ ) distorts the signed-distance field and underperforms the tanh baseline, whereas higher bandwidth ( $\sigma = 5$ ) is compatible with moderate regularization ( $w_\text{eik} = 10^{-3}$ ) and yields the global best result. To the best of our knowledge, this bandwidth-dependent constraint has not previously been identified in the PINN literature.
