The rapid growth of data in water resources has created new opportunities to accelerate knowledge discovery with the use of advanced deep learning tools. Hybrid models that integrate theory with state‐of‐the art empirical techniques have the potential to improve predictions while remaining true to physical laws. This paper evaluates the Process‐Guided Deep Learning (PGDL) hybrid modeling framework with a use‐case of predicting depth‐specific lake water temperatures. The PGDL model has three primary components: a deep learning model with temporal awareness (long short‐term memory recurrence), theory‐based feedback (model penalties for violating conversation of energy), and model pretraining to initialize the network with synthetic data (water temperature predictions from a process‐based model). In situ water temperatures were used to train the PGDL model, a deep learning (DL) model, and a process‐based (PB) model. Model performance was evaluated in various conditions, including when training data were sparse and when predictions were made outside of the range in the training data set. The PGDL model performance (as measured by root‐mean‐square error (RMSE)) was superior to DL and PB for two detailed study lakes, but only when pretraining data included greater variability than the training period. The PGDL model also performed well when extended to 68 lakes, with a median RMSE of 1.65 °C during the test period (DL: 1.78 °C, PB: 2.03 °C; in a small number of lakes PB or DL models were more accurate). This case‐study demonstrates that integrating scientific knowledge into deep learning tools shows promise for improving predictions of many important environmental variables.

This paper proposes a physics-guided recurrent neural network model (PGRNN) that combines RNNs and physicsbased 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. 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.

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.

We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter 0 < ε 1 which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength O(ε 2 ) and O(1) in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size h and time step τ as well as the small parameter ε. Based on the error bounds, in order to obtain 'correct' numerical solutions in the nonrelativistic limit regime, i.e. 0 < ε 1, the FDTD methods share the same ε-scalability on time step and mesh size as:. Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their ε-scalability is improved to τ = O(ε 2 ) and h = O(1) when 0 < ε 1. Extensive numerical results are reported to support our error estimates.

Super-resolution of the Lie-Trotter splitting (S 1 ) and Strang splitting (S 2 ) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic limit regime with a small parameter 0 < ε ≤ 1 inversely proportional to the speed of light. In this limit regime, the solution highly oscillates in time with wavelength at O(ε 2 ) in time. The splitting methods surprisingly show super-resolution, in the sense of breaking the resolution constraint under the Shannon's sampling theorem, i.e. the methods can capture the solution accurately even if the time step size τ is much larger than the sampled wavelength at O(ε 2 ). Similar to the linear case, S 1 and S 2 both exhibit 1/2 order convergence uniformly with respect to ε. Moreover, if τ is non-resonant, i.e. τ is away from certain region determined by ε, S 1 would yield an improved uniform first order O(τ ) error bound, while S 2 would give improved uniform 3/2 order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that super-resolution is still valid for higher order splitting methods.

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Physics-based models are often used to study engineering and environmental systems. The ability to model these systems is the key to achieving our future environmental sustainability and improving the quality of human life. This article focuses on simulating lake water temperature, which is critical for understanding the impact of changing climate on aquatic ecosystems and assisting in aquatic resource management decisions. General Lake Model (GLM) is a state-of-the-art physics-based model used for addressing such problems. However, like other physics-based models used for studying scientific and engineering systems, it has 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 article 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 (by over 20% even with very little training data), 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.

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