Weightsense: A Multifaceted Research Paradigm

• Weightsense is a term defining distinct technical mechanisms, including physical weight sensing, weight-driven learning, and sensitivity analysis in weighted estimators.
• In mobile and robotic contexts, it drives innovations such as smartwatch bite-weight estimation using inertial data and precise robotic grasping via explicit scale monitoring.
• In computational settings, weightsense informs methods like loss-weight selection in PINNs and sensitivity quantification in weighted least squares, enhancing system stability and inference accuracy.

Searching arXiv for papers on “weightsense” and the supplied identifiers to ground the article. {"^^^^^^^^7query7 OR \7^^^^^^^^"Estimation of Food Intake Quantity Using Inertial Signals from Smartwatches\"^^^^^^^^7 OR \7^^^^^^^^"CLAW: A Vision-Language-Action Framework for Weight-Aware Robotic Grasping\"^^^^^^^^7 OR \7^^^^^^^^"Digital Scale: Open-Source On-Device BMI Estimation from Smartphone Camera Images Trained on a Large-Scale Real-World Dataset\"^^^^^^^^7 OR \7^^^^^^^^"Weight-Based Representation Learning for Parameter Inference in Monte Carlo Simulations\"^^^^^^^^7 OR \7^^^^^^^^"Impact of Loss Weight and Model Complexity on Physics-Informed Neural Networks for Computational Fluid Dynamics\"^^^^^^^^7 OR \7^^^^^^^^"Sensitivity of weighted least squares estimators to omitted variables\"","max_results":^^^^7weightsense OR \7query7^^^^} WeightSense, or “weightsense,” is a polysemous research term used in recent arXiv literature to denote several technically distinct but structurally related ideas: estimation of physical weight or intake from wearable or vision signals, exploitation of simulator- or loss-derived weights as learning signals, and sensitivity analysis for weighted estimators. In smartwatch dietary monitoring, it refers to bite-weight estimation from wrist inertial micromovements; in robotic manipulation, to explicit scale reading for threshold-aware control; in mobile health, to BMI-based weight inference from smartphone images; in high-energy physics, to representation learning guided by event-level weights; in CFD PINNs, to principled loss-weight selection; and in statistics, to an R package for omitted-variable sensitivity in weighted least squares (&&&^^^^7query7^^^^&&&, &&&^^^^7weightsense OR \7^^^^&&&, &&&^^^^7 OR \7^^^^&&&, &&&^^^^7 OR \7^^^^&&&, &&&^^^^7 OR \7^^^^&&&, &&&^^^^7 OR \7^^^^&&&).

^^^^7weightsense OR \7^^^^. Terminological scope

Across these works, “weightsense” does not denote a single standardized method. Instead, it is used for different technical objects that share an emphasis on extracting, propagating, or auditing information carried by a notion of “weight,” whether that weight is physical mass, a simulator-provided event weight, a loss coefficient, or a regression weight (&&&^^^^7 OR \7^^^^&&&, &&&^^^^7 OR \7^^^^&&&).

Usage	Operational object	Representative papers
Physical-weight sensing	Bite mass, scale threshold, BMI-derived body weight	(&&&^^^^7query7^^^^&&&, &&&^^^^7weightsense OR \7^^^^&&&, &&&^^^^7 OR \7^^^^&&&)
Learning from weights	Event-level weights or loss weights	(&&&^^^^7 OR \7^^^^&&&, &&&^^^^7 OR \7^^^^&&&)
Sensitivity of weighted estimators	Weighted partial PRESERVED_PLACEHOLDER_^^^^7query7^^^^ and omitted-variable bias	(&&&^^^^7 OR \7^^^^&&&)

A plausible organizing principle is to divide the literature into three families. The first concerns direct sensing of food or body weight, or of a target mass during robotic manipulation. The second treats weights as auxiliary supervision or balancing variables inside a learning system. The third studies how substantive conclusions from weighted estimation change under omitted-variable sensitivity assumptions. The shared vocabulary is therefore functional rather than taxonomic.

^^^^7 OR \7^^^^. Smartwatch bite-weight estimation

In “Estimation of Food Intake Quantity Using Inertial Signals from Smartwatches” (&&&^^^^7query7^^^^&&&), weightsense refers to estimation of the grams of food consumed in each bite using only inertial signals from a commercial smartwatch. The study uses ^^^^7weightsense OR \7query7^^^^ adults recorded in semi-controlled, single-plate meals with conventional utensils. Signals come from the ^^^^7 OR \7^^^^D accelerometer and ^^^^7 OR \7^^^^D gyroscope of a Huawei Watch ^^^^7 OR \7^^^^ on the dominant wrist, while ground-truth bite weights are derived from a Bluetooth-enabled smart plate scale sampled at ^^^^7weightsense OR \7^^^^ Hz and bite intervals are manually annotated from GoPro Hero ^^^^7weightsense OR \7query7^^^^ video. The dataset contains ^^^^7 OR \7 OR \7 OR \7^^^^ total bites over approximately ^^^^7 OR \7^^^^.9 hours, with mean bite duration ^^^^7 OR \7^^^^.^^^^7 OR \7 OR \7^^^^ s, mean bite weight ^^^^7weightsense OR \7query7^^^^.89 g, and smartwatch IMU streams originally recorded at PRESERVED_PLACEHOLDER_^^^^7weightsense OR \7^^^^ Hz and resampled to ^^^^7weightsense OR \7query7query7^^^^ Hz via linear interpolation.

The preprocessing pipeline is explicitly engineered for wrist IMU stability. It resamples to ^^^^7weightsense OR \7query7query7^^^^ Hz, removes gravity from the accelerometer with a high-pass FIR filter with ^^^^7 OR \7query7weightsense OR \7^^^^ taps and ^^^^7weightsense OR \7^^^^ Hz cutoff applied forward–backward for zero-phase distortion, attenuates noise with a ^^^^7 OR \7^^^^th-order median filter, and mirrors left-hand recordings to right-hand orientation by inverting channels PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^, PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^, and PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^. Bite segmentation itself is not automatic in this study; it uses manually annotated start and end timestamps.

Feature design combines behavioral descriptors with statistical inertial summaries. The behavioral features are derived from temporal probabilities emitted by a previously trained micromovement classification model operating on ^^^^7query7^^^^.^^^^7 OR \7^^^^ s windows with ^^^^7query7^^^^.^^^^7weightsense OR \7^^^^ s step and predicting five gestures: pick food, upward movement, mouth, downward movement, and no movement. From these outputs, PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^ measures food-gathering duration and PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^ measures a stillness score during transport to the mouth, defined as

$$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$

The inertial features are computed from ^^^^7 OR \7^^^^ s sliding windows with ^^^^7query7^^^^.^^^^7weightsense OR \7^^^^ s step over the bite IMU matrix $M \in \mathbb{R}^{N \times 6}$, yielding $f_3$ through PRESERVED_PLACEHOLDER_^^^^7weightsense OR \7query7^^^^ as skewness, range, and entropy aggregates. The final bite descriptor is

PRESERVED_PLACEHOLDER_^^^^7weightsense OR \7weightsense OR \7^^^^

Estimation is performed with a linear SVR after z-score standardization. The selected hyperparameters are PRESERVED_PLACEHOLDER_^^^^7weightsense OR \7 OR \7^^^^ and PRESERVED_PLACEHOLDER_^^^^7weightsense OR \7 OR \7^^^^, evaluated under leave-one-subject-out cross-validation. The reported performance is a mean absolute error of ^^^^7 OR \7^^^^.99 g/bite and an improvement of ^^^^7weightsense OR \77^^^^.^^^^7 OR \7weightsense OR \7^^^^% relative to a baseline mean-weight predictor with MAE ^^^^7 OR \7^^^^.8^^^^7 OR \7^^^^ g/bite. The early-fusion deep model reaches MAE ^^^^7 OR \7^^^^.^^^^7 OR \77^^^^ g, MSE ^^^^7 OR \7query7^^^^.8^^^^7 OR \7^^^^ gPRESERVED_PLACEHOLDER_^^^^7weightsense OR \7 OR \7^^^^, and MAPE ^^^^7 OR \7^^^^8.^^^^7 OR \7 OR \7^^^^%, while the adapted state-of-the-art method under dominant-wrist IMU-only constraints performs worse than baseline, with MAE ^^^^7 OR \7^^^^.^^^^7 OR \7 OR \7^^^^ g, improvement PRESERVED_PLACEHOLDER_^^^^7weightsense OR \7 OR \7^^^^, MAPE 7^^^^7 OR \7^^^^.77%, and MSE ^^^^7 OR \7 OR \7^^^^.^^^^7query7 OR \7^^^^ gPRESERVED_PLACEHOLDER_^^^^7weightsense OR \7 OR \7^^^^. Subject-level variability remains substantial, with individual improvements ranging from ^^^^7 OR \7 OR \7^^^^.^^^^7 OR \7 OR \7^^^^% to PRESERVED_PLACEHOLDER_^^^^7weightsense OR \77^^^^, and meal-level aggregation gives a mean total meal difference of PRESERVED_PLACEHOLDER_^^^^7weightsense OR \7^^^^8 g. The paper accordingly presents feasibility rather than free-living deployment, noting the dependence on manual bite annotation, semi-controlled meals, dominant-wrist sensing, and a small cohort.

^^^^7 OR \7^^^^. Robotic weight-aware control through explicit scale monitoring

In “CLAW: A Vision-Language-Action Framework for Weight-Aware Robotic Grasping” (&&&^^^^7weightsense OR \7^^^^&&&), weightsense is realized by decoupling numeric weight monitoring from action generation. The paper argues that end-to-end vision-language-action models often fail to enforce precise numeric stopping conditions, even when a scale is visible, because they tend to internalize heuristics such as a learned number of grasps rather than an explicit threshold test. CLAW addresses this by using a fine-tuned CLIP model as a lightweight prompt generator and a flow-based VLA policy, PRESERVED_PLACEHOLDER_^^^^7weightsense OR \7^^^^9, as the action generator.

The scale-monitoring mechanism is binary. Given a scale crop and an instruction such as “load PRESERVED_PLACEHOLDER_^^^^7 OR \7query7^^^^ g target for me,” CLIP emits

PRESERVED_PLACEHOLDER_^^^^7 OR \7weightsense OR \7^^^^

with the runtime rule “continue if PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^; stop if PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^.” The action policy then conditions on PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ together with multi-view workspace images. The paper writes these conditionals as

PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^

The sensing stack uses three cameras for workspace observations and one fixed camera viewing the digital scale, with the numeric region cropped for CLIP. CLIP runs at ^^^^7 OR \7query7^^^^ Hz, while PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ runs at ^^^^7 OR \7query7^^^^ Hz and generates action chunks of ^^^^7 OR \7query7^^^^ time steps.

Training is split across the two modules. CLIP is fine-tuned on ^^^^7 OR \7query7query7query7^^^^ scale-display crops, each paired with PRESERVED_PLACEHOLDER_^^^^7 OR \77^^^^ synthetic instructions “load PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^8 g target for me.” for PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^9, producing PRESERVED_PLACEHOLDER_^^^^7 OR \7query7^^^^ samples with binary labels defined by the comparison between PRESERVED_PLACEHOLDER_^^^^7 OR \7weightsense OR \7^^^^ and the ground-truth scale reading. The flow-based policy is fine-tuned from ^^^^7 OR \7query7^^^^ demonstration episodes per task, with per-frame prompt labels that identify phase and, in mixed-object settings, object identity. Training proceeds for ^^^^7 OR \7query7^^^^,^^^^7query7query7query7^^^^ steps on an H^^^^7 OR \7query7query7^^^^ GPU.

Evaluation covers single-object candy or garlic tasks and mixed-object dual-arm setups at thresholds of ^^^^7 OR \7query7^^^^ g, ^^^^7 OR \7query7^^^^ g, and ^^^^7 OR \7query7^^^^ g. Raw-PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ action success ranges from ^^^^7query7^^^^.^^^^7weightsense OR \7 OR \7^^^^ to ^^^^7query7^^^^.^^^^7 OR \7 OR \7^^^^, while fine-tuned PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ and CLAW both achieve ^^^^7weightsense OR \7^^^^.^^^^7query7query7^^^^ action success across all tasks. The discriminating metric is stop-point success: raw-PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ is ^^^^7query7^^^^.^^^^7query7query7^^^^ or ^^^^7query7^^^^.^^^^7query7 OR \7^^^^, fine-tuned PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ ranges from ^^^^7query7^^^^.^^^^7query7query7^^^^ to ^^^^7query7^^^^.^^^^7 OR \7 OR \7^^^^, and CLAW achieves ^^^^7weightsense OR \7^^^^.^^^^7query7query7^^^^ across all candy and garlic thresholds. The paper attributes this to explicit sensing-to-prompt decoupling, high-frequency monitoring, and strong prompt sensitivity in PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^. Limitations remain tied to readable digital displays, fixed crop extraction, and potential degradation under lighting extremes or occlusions.

^^^^7 OR \7^^^^. Smartphone camera estimation of BMI and inferred body weight

In “Digital Scale: Open-Source On-Device BMI Estimation from Smartphone Camera Images Trained on a Large-Scale Real-World Dataset” (&&&^^^^7 OR \7^^^^&&&), weightsense refers to on-device estimation of BMI from a single smartphone image and subsequent conversion to body weight when height is known. The paper’s central empirical resource is the WayBED dataset: 8^^^^7 OR \7^^^^,9^^^^7 OR \7 OR \7^^^^ smartphone images from ^^^^7 OR \7 OR \7^^^^,^^^^7 OR \7 OR \7 OR \7^^^^ individuals, collected over more than ^^^^7weightsense OR \7query7^^^^ years in the WayBetter program, with self-reported height and verified weight through a weigh-in word and scale photo. Images are filtered by person detection, person-to-background ratio, and posture clustering based on ^^^^7weightsense OR \77^^^^ body keypoints; this removes ^^^^7weightsense OR \7 OR \7^^^^,^^^^7 OR \7 OR \7 OR \7^^^^ images and retains 7^^^^7weightsense OR \7^^^^,^^^^7 OR \7 OR \7 OR \7^^^^ high-quality samples.

The BMI estimator uses a DenseNet-^^^^7 OR \7query7weightsense OR \7^^^^ backbone with Squeeze-and-Excitation blocks after each transition layer. Training uses MSE on BMI, Adam with learning rate ^^^^7query7^^^^.^^^^7query7query7weightsense OR \7^^^^ and weight decay ^^^^7query7^^^^.^^^^7query7query7query7weightsense OR \7^^^^, batch size ^^^^7 OR \7 OR \7^^^^, and ^^^^7 OR \7query7^^^^ epochs, with learning-rate reduction by ^^^^7query7^^^^.^^^^7weightsense OR \7^^^^ after ^^^^7 OR \7^^^^ epochs of no validation improvement. Three perspectives are evaluated from the same full-body image: full-body, torso-up, and face-only. The full mobile pipeline is deployed on Android via CLAID; the BMI model is exported with ExecuTorch, while the filtering stack uses EfficientDet Lite^^^^7 OR \7^^^^ and MoveNet Thunder in TFLite.

Performance is reported on subject-disjoint splits. Full-body images achieve MAPE 7.9%, MAE ^^^^7 OR \7^^^^.^^^^7 OR \7 OR \7^^^^ BMI, and MAE 7.^^^^7 OR \7^^^^9 kg; torso-up gives MAPE 9.^^^^7weightsense OR \7^^^^%, MAE ^^^^7 OR \7^^^^.97 BMI, and MAE 8.^^^^7 OR \7weightsense OR \7^^^^ kg; face-only gives MAPE ^^^^7weightsense OR \7weightsense OR \7^^^^.^^^^7weightsense OR \7^^^^%, MAE ^^^^7 OR \7^^^^.^^^^7 OR \7 OR \7^^^^ BMI, and MAE ^^^^7weightsense OR \7query7^^^^.^^^^7 OR \7^^^^8 kg. Cross-dataset generalization from WayBED to unseen VisualBodyToBMI yields MAPE ^^^^7weightsense OR \7 OR \7^^^^.^^^^7 OR \7^^^^8%, while fine-tuning on VisualBodyToBMI reduces this to 8.^^^^7 OR \7 OR \7^^^^%, which the paper reports as the lowest value on that dataset so far.

The weight-sensing step is explicit rather than implicit. Using

PRESERVED_PLACEHOLDER_^^^^7 OR \77^^^^

the model’s BMI estimate becomes a weight estimate when height is provided. The paper also gives a first-order error propagation relation,

PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^8

For the WayBED hold-out, BMI MAPE of 7.9% combined with height error of about ^^^^7weightsense OR \7^^^^% suggests weight relative error of about 9.9%; for a true 8^^^^7query7^^^^ kg person, the example error is approximately PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^9 kg. The paper emphasizes telehealth and emergency scenarios, but also notes important limitations: single-image inference, no reference scale in the image, residual label bias from self-reported height, incomplete demographic reporting, absence of subgroup error analysis, and sensitivity to clothing, posture, occlusion, and camera geometry.

^^^^7 OR \7^^^^. Event-level weights as weak supervision in Monte Carlo physics

“Weight-Based Representation Learning for Parameter Inference in Monte Carlo Simulations” (&&&^^^^7 OR \7^^^^&&&) uses “weightsense” in a different sense: not physical mass sensing, but extraction of parameter sensitivity from simulator-provided event weights. The paper states that the authors do not explicitly coin a name like “WeightSense”; the term is introduced there as an explanatory shorthand for learning parameter-informative representations from event-level weights and then using those representations in likelihood-based inference.

The setting is simulator-based inference for the top Yukawa coupling, written primarily in terms of

PRESERVED_PLACEHOLDER_^^^^7 OR \7query7^^^^

Each simulated event is accompanied by weights that encode how its probability or yield changes with the model parameter. The method defines weak labels for four-top-quark signal events by comparing weight variation across sampled PRESERVED_PLACEHOLDER_^^^^7 OR \7weightsense OR \7^^^^ points, producing “low-weight” and “high-weight” classes. A parameter-inference network with ^^^^7 OR \7 OR \7^^^^ input features, BatchNorm, dense tanh layers with dropout, and a sigmoid output is trained with binary cross-entropy on these weak labels. A separate background-rejection network with ^^^^7 OR \7 OR \7^^^^ input features, BatchNorm, dense ReLU layers with dropout, and a ^^^^7 OR \7^^^^-node softmax output separates PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^, PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^, and PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^.

The learned outputs are discretized in two stages. First, the background-network output space is partitioned into ^^^^7 OR \7 OR \7^^^^ tiles on a ^^^^7query7^^^^.^^^^7weightsense OR \7^^^^ by ^^^^7query7^^^^.^^^^7weightsense OR \7^^^^ grid over the two background-node axes. Then, within each tile, the parameter-network output is binned into up to six bins using a split-and-equalize procedure, with bin merging if necessary. This produces the final template for a binned Poisson likelihood,

PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^

Yields are parameterized by reweighted sums of event weights: the four-top signal uses a fourth-order polynomial in PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^, the PRESERVED_PLACEHOLDER_^^^^7 OR \77^^^^ background a second-order polynomial, and PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^8 scales as PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^9.

In the direct parameterized-yield setting, the reported ^^^^7 OR \7^^^^8% confidence intervals are PRESERVED_PLACEHOLDER_^^^^7 OR \7query7^^^^ for ^^^^7 OR \7query7weightsense OR \77^^^^ CMS-like data, PRESERVED_PLACEHOLDER_^^^^7 OR \7weightsense OR \7^^^^ for ^^^^7 OR \7query7weightsense OR \7 OR \7^^^^–^^^^7 OR \7query7weightsense OR \7^^^^8 Run ^^^^7 OR \7^^^^, and PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ for the HL-LHC scenario. In an alternative configuration where PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ is fixed at PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ and PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ normalization floats freely, the method yields upper bounds only: at ^^^^7 OR \7^^^^8% and 9^^^^7 OR \7^^^^% CL, ^^^^7 OR \7query7weightsense OR \77^^^^ gives ^^^^7 OR \7^^^^.^^^^7 OR \7query7^^^^8 and ^^^^7 OR \7^^^^.8^^^^7query77^^^^, Run ^^^^7 OR \7^^^^ gives ^^^^7 OR \7^^^^.^^^^7query7 OR \7 OR \7^^^^ and ^^^^7 OR \7^^^^.^^^^7 OR \7 OR \7 OR \7^^^^, and HL-LHC gives ^^^^7weightsense OR \7^^^^.^^^^7 OR \7 OR \7 OR \7^^^^ and ^^^^7 OR \7^^^^.^^^^7query7weightsense OR \7 OR \7^^^^. The paper reports that direct weight-based inference gives tighter upper bounds than an indirect cross-section-surrogate translation to PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^, and also extends the same summary statistics to inference in the PRESERVED_PLACEHOLDER_^^^^7 OR \77^^^^ plane for CP-even and CP-odd couplings. The main caveats concern weight fidelity, sparse regions, possible instability from negative weights, and the growing importance of systematic uncertainties at high luminosity.

^^^^7 OR \7^^^^. Loss-weight selection in physics-informed neural networks

In “Impact of Loss Weight and Model Complexity on Physics-Informed Neural Networks for Computational Fluid Dynamics” (&&&^^^^7 OR \7^^^^&&&), weightsense is defined as the capability to select and adapt loss weights so that PINN training remains balanced across PDE residuals, boundary or initial conditions, and data-fidelity terms. The basic objective is

PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^8

The paper argues that without principled weighting, gradients are dominated by the largest-magnitude component, leading to biased updates, slow or stalled convergence, and nonphysical solutions.

Two weighting schemes are proposed. The first is a strict dimensional-analysis-based balancing scheme, denoted by the subscript “NMPRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^9,” which makes each weighted component contribute on the same order of magnitude. The second is a relaxed scheme, denoted “NM,” that takes square roots of the quantifiable ratios to account for “unquantifiable” contributions such as data noise, boundary layers, corner singularities, sampling variability, and network-induced stiffness. The relaxed weights used in the experiments are:

• conduction: PRESERVED_PLACEHOLDER_^^^^7 OR \7query7^^^^;
• convection–diffusion: PRESERVED_PLACEHOLDER_^^^^7 OR \7weightsense OR \7^^^^;
• lid-driven cavity: PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^.

The networks are fully connected with sinusoidal activations. Scalar-output PINNs use ^^^^7 OR \7^^^^ hidden layers with ^^^^7 OR \7 OR \7^^^^ neurons each, while vector-output PINNs use ^^^^7 OR \7^^^^ hidden layers with widths [^^^^7 OR \7 OR \7^^^^, ^^^^7 OR \7query7^^^^, ^^^^7 OR \7query7^^^^, ^^^^7 OR \7query7^^^^]. Residuals are evaluated at collocation points via central differencing with spacing PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^, and experiments cover heat conduction, convection–diffusion, and lid-driven cavity flow.

The numerical results position weightsense here as a training-stability mechanism. For heat conduction at PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^, MSE PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^ is ^^^^7 OR \7^^^^.7^^^^7 OR \7 OR \7^^^^ with equal weighting, ^^^^7 OR \7^^^^.^^^^7 OR \7 OR \77^^^^ with strict order balancing, and ^^^^7query7^^^^.^^^^7weightsense OR \7^^^^9^^^^7 OR \7^^^^ with the relaxed scheme. For convection–diffusion at PRESERVED_PLACEHOLDER_^^^^7 OR \7 OR \7^^^^, equal weighting fails at PRESERVED_PLACEHOLDER_^^^^7 OR \77^^^^ with ^^^^7weightsense OR \7 OR \7 OR \7^^^^.7 and at PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^8 with ^^^^7 OR \7weightsense OR \7^^^^8.^^^^7 OR \7^^^^, whereas the relaxed scheme gives ^^^^7query7^^^^.^^^^7 OR \7max_results7 OR \7^^^^ and ^^^^7weightsense OR \7^^^^.^^^^7query7max_results7 OR \7^^^^, and strict gives ^^^^7weightsense OR \7^^^^.^^^^7weightsense OR \7 OR \7 OR \7^^^^ and ^^^^7weightsense OR \7^^^^.^^^^7weightsense OR \7 OR \7 OR \7^^^^. At PRESERVED_PLACEHOLDER_^^^^7 OR \7^^^^9, equal weighting fails across resolutions with errors from ^^^^7weightsense OR \7 OR \7 OR \7^^^^.^^^^7weightsense OR \7^^^^ to ^^^^7 OR \7 OR \7 OR \7^^^^.^^^^7query7^^^^, while relaxed and strict remain low; at $$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$^^^^7query7^^^^, relaxed gives ^^^^7query7^^^^.7^^^^7 OR \7 OR \7^^^^ and strict ^^^^7query7^^^^.7^^^^7query7 For lid-driven cavity at $$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$^^^^7weightsense OR \7^^^^ and $$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$^^^^7 OR \7^^^^, strict degrades to ^^^^7weightsense OR \7weightsense OR \7^^^^.^^^^7query77^^^^, whereas equal gives ^^^^7 OR \7^^^^.^^^^7 OR \7 OR \7 OR \7^^^^ and relaxed ^^^^7 OR \7^^^^.^^^^7query7^^^^88. The paper therefore treats relaxed dimensional weighting as the practical default, particularly in high-Peclet-number regimes where equal weighting fails.

7. Sensitivity analysis for weighted least squares estimators

In “Sensitivity of weighted least squares estimators to omitted variables” (&&&^^^^7 OR \7^^^^&&&), weightsense is an R package implementing omitted-variable sensitivity analysis for weighted linear regression of an outcome on a treatment and observed covariates. The weighted least squares estimator is written as

$$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$^^^^7 OR \7^^^^

where $$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$^^^^7 OR \7^^^^ and $$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$^^^^7 OR \7^^^^. The package’s central methodological choice is to hold the weights fixed and study how the weighted outcome model changes if an unobserved confounder $$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$^^^^7 OR \7^^^^ were added.

Sensitivity is parameterized through two weighted partial $$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$7 quantities. The first is a treatment-side parameter, the proportion of weighted residual variance in $$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$8 given $$f_2 = (1 - V_{\text{norm}}) + \log(D_{\text{norm}} + 1).$$9 explained by $M \in \mathbb{R}^{N \times 6}$^^^^7query7^^^^. The second is an outcome-side parameter, the proportion of weighted residual variance in $M \in \mathbb{R}^{N \times 6}$^^^^7weightsense OR \7^^^^ given $M \in \mathbb{R}^{N \times 6}$^^^^7 OR \7^^^^ and $M \in \mathbb{R}^{N \times 6}$^^^^7 OR \7^^^^ explained by $M \in \mathbb{R}^{N \times 6}$^^^^7 OR \7^^^^. These parameters are bounded between ^^^^7query7^^^^ and ^^^^7weightsense OR \7^^^^ and feed directly into an omitted-variable bias expression for the WLS coefficient on $M \in \mathbb{R}^{N \times 6}$^^^^7 OR \7^^^^. From this, the paper develops robustness values, an extreme-scenario diagnostic, benchmarking bounds against observed covariates via semi-weights, and bootstrap-based adjusted inference. The framework is designed to work with inverse propensity score, matching, covariate balancing, and stratification weights, and it does so without imposing distributional assumptions on the data or on $M \in \mathbb{R}^{N \times 6}$^^^^7 OR \7^^^^.

Benchmarking is the paper’s distinctive weighted extension. Because balancing weights can nearly destroy the weighted association between $M \in \mathbb{R}^{N \times 6}$7 and a benchmark covariate $M \in \mathbb{R}^{N \times 6}$8, treatment-side calibration uses semi-weights $M \in \mathbb{R}^{N \times 6}$9 obtained by recomputing the weighting scheme while omitting $f_3$^^^^7query7^^^^. This yields relative-strength parameters comparing $f_3$^^^^7weightsense OR \7^^^^ to observed covariates in their relation to treatment and outcome. The package also supports percentile bootstrap, cluster bootstrap, and a fixed-weight bootstrap variant that is especially relevant for matching with replacement.

The applied illustration revisits Hazlett’s Darfur study under several weighting schemes. For IPW targeting the ATE, the estimate is ^^^^7query7^^^^.^^^^7query7 with 9^^^^7 OR \7^^^^% CI (^^^^7query7^^^^.^^^^7query7 OR \7 OR \7^^^^, ^^^^7query7^^^^.^^^^7weightsense OR \7 OR \7^^^^8), $f_3$^^^^7 OR \7^^^^, and benchmarking “$f_3$^^^^7 OR \7^^^^ as strong as Female” gives adjusted estimate ^^^^7query7^^^^.^^^^7query7 OR \7^^^^9 with 9^^^^7 OR \7^^^^% CI (^^^^7query7^^^^.^^^^7query7weightsense OR \7 OR \7^^^^, ^^^^7query7^^^^.^^^^7weightsense OR \7weightsense OR \77^^^^). For propensity score matching for the ATT, the estimate is ^^^^7query7^^^^.^^^^7query77 with 9^^^^7 OR \7^^^^% CI (^^^^7query7^^^^.^^^^7query7 OR \7weightsense OR \7^^^^, ^^^^7query7^^^^.^^^^7weightsense OR \7 OR \7weightsense OR \7^^^^), and under the benchmark “$f_3$^^^^7 OR \7^^^^ twice as strong as Female and Age for $f_3$^^^^7 OR \7^^^^, equally strong for $f_3$^^^^7 OR \7^^^^,” the adjusted estimate becomes ^^^^7query7^^^^.^^^^7query7 OR \7^^^^8 with 9^^^^7 OR \7^^^^% CI (-^^^^7query7^^^^.^^^^7query7query7 OR \7^^^^, ^^^^7query7^^^^.^^^^7weightsense OR \7 OR \7 OR \7^^^^). For entropy balancing on gender and village for the ATT, the estimate is ^^^^7query7^^^^.^^^^7query7 OR \7^^^^ with 9^^^^7 OR \7^^^^% CI (^^^^7query7^^^^.^^^^7query7 OR \7^^^^9, ^^^^7query7^^^^.^^^^7weightsense OR \7 OR \7query7^^^^), and benchmarking “$f_3$7 as strong as Female” yields adjusted estimate ^^^^7query7^^^^.^^^^7query7max_results7 OR \7^^^^ with 9^^^^7 OR \7^^^^% CI (^^^^7query7^^^^.^^^^7query7 OR \7 OR \7^^^^, ^^^^7query7^^^^.^^^^7weightsense OR \7 OR \7 OR \7^^^^). The paper concludes that results are generally robust to confounding as strong as gender, while stronger treatment-side confounding assumptions are needed to overturn conclusions in more fragile matching scenarios.

Across these domains, weightsense functions less as a single theory than as a recurring design pattern: identify what “weight” encodes in a given system, then build sensing, learning, optimization, or inferential machinery that makes that information operational. In some contexts the relevant object is literal mass; in others it is a simulator reweighting factor, a loss coefficient, or a regression weight. The term’s unifying significance lies in this transfer of informational burden onto weight-structured quantities, but the mathematical objects, assumptions, and evaluation criteria remain domain-specific.