Jared Willard

Ankur Mahesh, William Collins, Boris Bonev et al.

Studying low-likelihood high-impact extreme weather events in a warming world is a significant and challenging task for current ensemble forecasting systems. While these systems presently use up to 100 members, larger ensembles could enrich the sampling of internal variability. They may capture the long tails associated with climate hazards better than traditional ensemble sizes. Due to computational constraints, it is infeasible to generate huge ensembles (comprised of 1,000-10,000 members) with traditional, physics-based numerical models. In this two-part paper, we replace traditional numerical simulations with machine learning (ML) to generate hindcasts of huge ensembles. In Part I, we construct an ensemble weather forecasting system based on Spherical Fourier Neural Operators (SFNO), and we discuss important design decisions for constructing such an ensemble. The ensemble represents model uncertainty through perturbed-parameter techniques, and it represents initial condition uncertainty through bred vectors, which sample the fastest growing modes of the forecast. Using the European Centre for Medium-Range Weather Forecasts Integrated Forecasting System (IFS) as a baseline, we develop an evaluation pipeline composed of mean, spectral, and extreme diagnostics. Using large-scale, distributed SFNOs with 1.1 billion learned parameters, we achieve calibrated probabilistic forecasts. As the trajectories of the individual members diverge, the ML ensemble mean spectra degrade with lead time, consistent with physical expectations. However, the individual ensemble members' spectra stay constant with lead time. Therefore, these members simulate realistic weather states, and the ML ensemble thus passes a crucial spectral test in the literature. The IFS and ML ensembles have similar Extreme Forecast Indices, and we show that the ML extreme weather forecasts are reliable and discriminating.

Ankur Mahesh, William Collins, Boris Bonev et al.

In Part I, we created an ensemble based on Spherical Fourier Neural Operators. As initial condition perturbations, we used bred vectors, and as model perturbations, we used multiple checkpoints trained independently from scratch. Based on diagnostics that assess the ensemble's physical fidelity, our ensemble has comparable performance to operational weather forecasting systems. However, it requires orders of magnitude fewer computational resources. Here in Part II, we generate a huge ensemble (HENS), with 7,424 members initialized each day of summer 2023. We enumerate the technical requirements for running huge ensembles at this scale. HENS precisely samples the tails of the forecast distribution and presents a detailed sampling of internal variability. HENS has two primary applications: (1) as a large dataset with which to study the statistics and drivers of extreme weather and (2) as a weather forecasting system. For extreme climate statistics, HENS samples events 4$σ$ away from the ensemble mean. At each grid cell, HENS increases the skill of the most accurate ensemble member and enhances coverage of possible future trajectories. As a weather forecasting model, HENS issues extreme weather forecasts with better uncertainty quantification. It also reduces the probability of outlier events, in which the verification value lies outside the ensemble forecast distribution.

Shaoming Xu, Ankush Khandelwal, Xiang Li et al.

In many environmental applications, recurrent neural networks (RNNs) are often used to model physical variables with long temporal dependencies. However, due to mini-batch training, temporal relationships between training segments within the batch (intra-batch) as well as between batches (inter-batch) are not considered, which can lead to limited performance. Stateful RNNs aim to address this issue by passing hidden states between batches. Since Stateful RNNs ignore intra-batch temporal dependency, there exists a trade-off between training stability and capturing temporal dependency. In this paper, we provide a quantitative comparison of different Stateful RNN modeling strategies, and propose two strategies to enforce both intra- and inter-batch temporal dependency. First, we extend Stateful RNNs by defining a batch as a temporally ordered set of training segments, which enables intra-batch sharing of temporal information. While this approach significantly improves the performance, it leads to much larger training times due to highly sequential training. To address this issue, we further propose a new strategy which augments a training segment with an initial value of the target variable from the timestep right before the starting of the training segment. In other words, we provide an initial value of the target variable as additional input so that the network can focus on learning changes relative to that initial value. By using this strategy, samples can be passed in any order (mini-batch training) which significantly reduces the training time while maintaining the performance. In demonstrating our approach in hydrological modeling, we observe that the most significant gains in predictive accuracy occur when these methods are applied to state variables whose values change more slowly, such as soil water and snowpack, rather than continuously moving flux variables such as streamflow.

Jared D. Willard, Charuleka Varadharajan, Xiaowei Jia et al.

Prediction of dynamic environmental variables in unmonitored sites remains a long-standing challenge for water resources science. The majority of the world's freshwater resources have inadequate monitoring of critical environmental variables needed for management. Yet, the need to have widespread predictions of hydrological variables such as river flow and water quality has become increasingly urgent due to climate and land use change over the past decades, and their associated impacts on water resources. Modern machine learning methods increasingly outperform their process-based and empirical model counterparts for hydrologic time series prediction with their ability to extract information from large, diverse data sets. We review relevant state-of-the art applications of machine learning for streamflow, water quality, and other water resources prediction and discuss opportunities to improve the use of machine learning with emerging methods for incorporating watershed characteristics into deep learning models, transfer learning, and incorporating process knowledge into machine learning models. The analysis here suggests most prior efforts have been focused on deep learning learning frameworks built on many sites for predictions at daily time scales in the United States, but that comparisons between different classes of machine learning methods are few and inadequate. We identify several open questions for time series predictions in unmonitored sites that include incorporating dynamic inputs and site characteristics, mechanistic understanding and spatial context, and explainable AI techniques in modern machine learning frameworks.

Jared D. Willard, Peter Harrington, Shashank Subramanian et al.

The rapid rise of deep learning (DL) in numerical weather prediction (NWP) has led to a proliferation of models which forecast atmospheric variables with comparable or superior skill than traditional physics-based NWP. However, among these leading DL models, there is a wide variance in both the training settings and architecture used. Further, the lack of thorough ablation studies makes it hard to discern which components are most critical to success. In this work, we show that it is possible to attain high forecast skill even with relatively off-the-shelf architectures, simple training procedures, and moderate compute budgets. Specifically, we train a minimally modified SwinV2 transformer on ERA5 data, and find that it attains superior forecast skill when compared against IFS. We present some ablations on key aspects of the training pipeline, exploring different loss functions, model sizes and depths, and multi-step fine-tuning to investigate their effect. We also examine the model performance with metrics beyond the typical ACC and RMSE, and investigate how the performance scales with model size.

Runlong Yu, Shengyu Chen, Yiqun Xie et al.

Modeling environmental ecosystems is essential for effective resource management, sustainable development, and understanding complex ecological processes. However, traditional data-driven methods face challenges in capturing inherently complex and interconnected processes and are further constrained by limited observational data in many environmental applications. Foundation models, which leverages large-scale pre-training and universal representations of complex and heterogeneous data, offer transformative opportunities for capturing spatiotemporal dynamics and dependencies in environmental processes, and facilitate adaptation to a broad range of applications. This survey presents a comprehensive overview of foundation model applications in environmental science, highlighting advancements in common environmental use cases including forward prediction, data generation, data assimilation, downscaling, inverse modeling, model ensembling, and decision-making across domains. We also detail the process of developing these models, covering data collection, architecture design, training, tuning, and evaluation. Through discussions on these emerging methods as well as their future opportunities, we aim to promote interdisciplinary collaboration that accelerates advancements in machine learning for driving scientific discovery in addressing critical environmental challenges.

Jared D. Willard, Fabio Ciulla, Helen Weierbach et al.

The prediction of streamflows and other environmental variables in unmonitored basins is a grand challenge in hydrology. Recent machine learning (ML) models can harness vast datasets for accurate predictions at large spatial scales. However, there are open questions regarding model design and data needed for inputs and training to improve performance. This study explores these questions while demonstrating the ability of deep learning models to make accurate stream temperature predictions in unmonitored basins across the conterminous United States. First, we compare top-down models that utilize data from a large number of basins with bottom-up methods that transfer ML models built on local sites, reflecting traditional regionalization techniques. We also evaluate an intermediary grouped modeling approach that categorizes sites based on regional co-location or similarity of catchment characteristics. Second, we evaluate trade-offs between model complexity, prediction accuracy, and applicability for more target locations by systematically removing inputs. We then examine model performance when additional training data becomes available due to reductions in input requirements. Our results suggest that top-down models significantly outperform bottom-up and grouped models. Moreover, it is possible to get acceptable accuracy by reducing both dynamic and static inputs enabling predictions for more sites with lower model complexity and computational needs. From detailed error analysis, we determined that the models are more accurate for sites primarily controlled by air temperatures compared to locations impacted by groundwater and dams. By addressing these questions, this research offers a comprehensive perspective on optimizing ML model design for accurate predictions in unmonitored regions.

Jared D. Willard, Jordan S. Read, Alison P. Appling et al.

Most environmental data come from a minority of well-monitored sites. An ongoing challenge in the environmental sciences is transferring knowledge from monitored sites to unmonitored sites. Here, we demonstrate a novel transfer learning framework that accurately predicts depth-specific temperature in unmonitored lakes (targets) by borrowing models from well-monitored lakes (sources). This method, Meta Transfer Learning (MTL), builds a meta-learning model to predict transfer performance from candidate source models to targets using lake attributes and candidates' past performance. We constructed source models at 145 well-monitored lakes using calibrated process-based modeling (PB) and a recently developed approach called process-guided deep learning (PGDL). We applied MTL to either PB or PGDL source models (PB-MTL or PGDL-MTL, respectively) to predict temperatures in 305 target lakes treated as unmonitored in the Upper Midwestern United States. We show significantly improved performance relative to the uncalibrated process-based General Lake Model, where the median RMSE for the target lakes is $2.52^{\circ}C$. PB-MTL yielded a median RMSE of $2.43^{\circ}C$; PGDL-MTL yielded $2.16^{\circ}C$; and a PGDL-MTL ensemble of nine sources per target yielded $1.88^{\circ}C$. For sparsely monitored target lakes, PGDL-MTL often outperformed PGDL models trained on the target lakes themselves. Differences in maximum depth between the source and target were consistently the most important predictors. Our approach readily scales to thousands of lakes in the Midwestern United States, demonstrating that MTL with meaningful predictor variables and high-quality source models is a promising approach for many kinds of unmonitored systems and environmental variables.

Xiaowei Jia, Jacob Zwart, Jeffrey Sadler et al.

This paper proposes a physics-guided machine learning approach that combines advanced machine learning models and physics-based models to improve the prediction of water flow and temperature in river networks. We first build a recurrent graph network model to capture the interactions among multiple segments in the river network. Then we present a pre-training technique which transfers knowledge from physics-based models to initialize the machine learning model and learn the physics of streamflow and thermodynamics. We also propose a new loss function that balances the performance over different river segments. We demonstrate the effectiveness of the proposed method in predicting temperature and streamflow in a subset of the Delaware River Basin. In particular, we show that the proposed method brings a 33\%/14\% improvement over the state-of-the-art physics-based model and 24\%/14\% over traditional machine learning models (e.g., Long-Short Term Memory Neural Network) in temperature/streamflow prediction using very sparse (0.1\%) observation data for training. The proposed method has also been shown to produce better performance when generalized to different seasons or river segments with different streamflow ranges.

Jared Willard, Xiaowei Jia, Shaoming Xu et al.

There is a growing consensus that solutions to complex science and engineering problems require novel methodologies that are able to integrate traditional physics-based modeling approaches with state-of-the-art machine learning (ML) techniques. This paper provides a structured overview of such techniques. Application-centric objective areas for which these approaches have been applied are summarized, and then classes of methodologies used to construct physics-guided ML models and hybrid physics-ML frameworks are described. We then provide a taxonomy of these existing techniques, which uncovers knowledge gaps and potential crossovers of methods between disciplines that can serve as ideas for future research.

Xiaowei Jia, Jared Willard, Anuj Karpatne et al.

Physics-based models of dynamical systems are often used to study engineering and environmental systems. Despite their extensive use, these models have several well-known limitations due to simplified representations of the physical processes being modeled or challenges in selecting appropriate parameters. While-state-of-the-art machine learning models can sometimes outperform physics-based models given ample amount of training data, they can produce results that are physically inconsistent. This paper proposes a physics-guided recurrent neural network model (PGRNN) that combines RNNs and physics-based models to leverage their complementary strengths and improves the modeling of physical processes. Specifically, we show that a PGRNN can improve prediction accuracy over that of physics-based models, while generating outputs consistent with physical laws. An important aspect of our PGRNN approach lies in its ability to incorporate the knowledge encoded in physics-based models. This allows training the PGRNN model using very few true observed data while also ensuring high prediction accuracy. Although we present and evaluate this methodology in the context of modeling the dynamics of temperature in lakes, it is applicable more widely to a range of scientific and engineering disciplines where physics-based (also known as mechanistic) models are used, e.g., climate science, materials science, computational chemistry, and biomedicine.

Xiaowei Jia, Jared Willard, Anuj Karpatne et al.

This paper proposes a physics-guided recurrent neural network model (PGRNN) that combines RNNs and physics-based models to leverage their complementary strengths and improve the modeling of physical processes. Specifically, we show that a PGRNN can improve prediction accuracy over that of physical models, while generating outputs consistent with physical laws, and achieving good generalizability. Standard RNNs, even when producing superior prediction accuracy, often produce physically inconsistent results and lack generalizability. We further enhance this approach by using a pre-training method that leverages the simulated data from a physics-based model to address the scarcity of observed data. The PGRNN has the flexibility to incorporate additional physical constraints and we incorporate a density-depth relationship. Both enhancements further improve PGRNN performance. Although we present and evaluate this methodology in the context of modeling the dynamics of temperature in lakes, it is applicable more widely to a range of scientific and engineering disciplines where mechanistic (also known as process-based) models are used, e.g., power engineering, climate science, materials science, computational chemistry, and biomedicine.

Xiaowei Jia, Anuj Karpatne, Jared Willard et al.

In this paper, we introduce a novel framework for combining scientific knowledge within physics-based models and recurrent neural networks to advance scientific discovery in many dynamical systems. We will first describe the use of outputs from physics-based models in learning a hybrid-physics-data model. Then, we further incorporate physical knowledge in real-world dynamical systems as additional constraints for training recurrent neural networks. We will apply this approach on modeling lake temperature and quality where we take into account the physical constraints along both the depth dimension and time dimension. By using scientific knowledge to guide the construction and learning the data-driven model, we demonstrate that this method can achieve better prediction accuracy as well as scientific consistency of results.