When Proxy Prediction Becomes Equation Reconstruction: Diagnostics and Residual Learning for Factor-Derived Proxy Supervision

Estimating temporal soil-loss change is challenging when physically meaningful input factors are noisy or corrupted, particularly because substantial changes are rare relative to the large number of locations exhibiting little change. We study this problem through the Revised Universal Soil Loss Equation (RUSLE) and introduce PhyRestore, a physics-structured latent-factor restoration framework. Rather than directly predicting soil-loss change or correcting a degraded physical estimate, PhyRestore restores corrupted physical factors and reconstructs temporal change through the known physical relationship. We evaluate PhyRestore in a watershed-scale bitemporal raster setting under isolated and simultaneous corruption of rainfall erosivity and cover management, comparing it with the degraded RUSLE estimate and Direct RF, XGBoost, MLP, and CNN models. Factor restoration improves high-magnitude recovery when the corrupted factors remain identifiable, but its advantage weakens under joint corruption, sparse positive extremes, and factor values outside the training support.

Data-driven remaining useful life (RUL) prediction requires complete degradation trajectories for training, yet such run-to-failure data are scarce and expensive. Practitioners currently lack principled guidance on how many failure examples suffice for a given model and accuracy target. This paper develops a sample complexity framework for RUL prediction comprising seven main results organised around three themes. First, we establish fundamental learning rates: a distribution-free generalization bound shows that the uniform deviation of the mean squared error decreases as $O(B^{2}\sqrt{p/n})$, where $p$ is the model complexity and $n$ the number of trajectories, and a minimax lower bound proves that the $Θ(p/n)$ rate is unimprovable.} \rev{Second, we quantify how domain knowledge accelerates learning: incorporating degradation physics reduces data requirements by up to two orders of magnitude for deep networks, a Bernstein-type analysis achieves the minimax-optimal $O(p/n)$ rate under high signal-to-noise conditions, and closed-form penalties reveal when an incorrectly assumed physics model hurts rather than helps. Third, we characterise the impact of data quality: fleet variability induces an irreducible bias$-$variance tradeoff, while right-censored observations suffer an efficiency loss that depends critically on the degradation class.} Closed-form expressions are provided for exponential, power-law, and stretched-exponential degradation. \rev{Cross-domain validation against published turbofan, battery, and bearing benchmarks confirms the theoretical predictions within a factor of 2$-$3 on average. The results yield practical guidelines for planning data collection, selecting model complexity, and evaluating physics model assumptions in prognostics applications.

Supervised learning with Earth observation inputs is often limited by the sparsity of high-quality labeled or in-situ measured data to use as training labels. With the abundance of geographic data products, in many cases there are variables correlated with - but different from - the variable of interest that can be leveraged. We integrate such proxy variables within a geographic prior via a trainable location encoder and introduce a proxy consistency loss (PCL) formulation to imbue proxy data into the location encoder. The first key insight behind our approach is to use the location encoder as an agile and flexible way to learn from abundantly available proxy data which can be sampled independently of training label availability. Our second key insight is that we will need to regularize the location encoder appropriately to achieve performance and robustness with limited labeled data. Our experiments on air quality prediction and poverty mapping show that integrating proxy data implicitly through the location encoder outperforms using both as input to an observation encoder and fusion strategies that use frozen, pretrained location embeddings as a geographic prior. Superior performance for in-sample prediction shows that the PCL can incorporate rich information from the proxies, and superior out-of-sample prediction shows that the learned latent embeddings help generalize to areas without training labels.