We introduce an algorithm for combining two sources of similar data within a loss function to use in an optimisation procedure. The technique involves training model parameters on a ratio of two data sources to encourage convergence on meta-model parameters that map behaviours learned from each data source. The model can achieve higher quality results when unknown error distorts the signal of either source. Using a differentiable loss function we use gradient descent to optimise the hyperparameter which controls the ratio of data signals while training the model weights. We derive the theoretical bounds of the loss function based on the potential error of the meta-model, and the loss function is numerically interpreted to show the solution space. To utilise the methodology, we show its use in an example optimisation algorithm. Using this algorithm, we conducted an experimental case study to train functionapproximating surrogate models of the simple diffusion equation and the nonlinear Schrödinger equation. Thus, we show the ratio-coupled loss function may be utilised in approximating linear and nonlinear partial differential equations.

Reliable oceanic and climate analysis depend on high-quality sensor readings, yet these systems commonly encounter significant sensor limitations, leading to missing data. Addressing this issue is critical for ensuring accurate forecasts and analyses. In this work, the data gap problem is studied by developing physics-regularized machine learning models with multiple-location modeling to forecast missing sensor data. Utilized are recurrent statistical surrogate models that generate hourly 24-hour forecasts. To train these models, we use a selection of five sensor features collected over three years. Introduced is a multi-location modeling scheme that uniquely combines sensor data from nearby buoys as a novel methodology. This approach allows for more stable and accurate predictions compared to forecasting with single buoy data alone. Our experiments reveal that grouping six buoys yields the best forecasting performance. Furthermore, we improve model accuracy by integrating buoy data with numerical ocean models and applying a physicsregularized loss function. This technique mitigates the impact of missing or erratic data, leading to more dependable 24hour forecasts. Our findings demonstrate that the combination of multiple-location modeling and physics-based regularization enhances the stability and accuracy of oceanic data forecasting.

One challenge in oceanographic analysis is the need for accurate initial conditions collected from physical buoys. Temporary sensor outages or noisy conditions can hinder the data collection process. Machine learning surrogate models offer short-term coverage during outages. This study presents a methodology for regularizing machine learning models that predict buoy observations by utilizing multiple data sources. A previous work introduced a ratio-coupling hyperparameter to combine numerically modeled data and ocean observations when calculating training loss. However, applying one ratio across all features failed to capture the unique characteristics of different data sources. To overcome this limitation, this work investigates a multiple-hyperparameter loss function to independently manage the contribution of each data source per feature. A bounded random grid search explores the hyperparameter space to find ratios which produce superior results compared to the singleratio approach. Surrogate models are validated at the same 88 fixed locations as the previous paper for a direct comparison. The experimental results suggest that this multi-ratio methodology can offer more reliable forecasts over a 24-hour period by applying the correct weight for each pairing of observed feature and numerical model source.

Methodologies inspired by physics-informed neural networks (PINNs) were used to forecast observations recorded by stationary ocean buoys. We combined buoy observations with numerical models to train surrogate deep learning networks that performed better than with either data alone. Numerical model outputs were collected from two sources for training and regularization: the hybrid circulation ocean model and the fifth ECMWF reanalysis experiment. A hyperparameter determines the ratio of observational and modeled data to be used in the training procedure, so we conducted a grid search to find the most performant ratio. Overall,  the technique improved the general forecast performance compared with nonregularized models. Under specific circumstances, the regularization mechanism enabled the PINN models to be more accurate than the numerical models. This demonstrates the utility of combining various climate models and sensor observations to improve surrogate modeling.